#72285403

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#15258
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
LCS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

4519 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(17,18)(19,27)(20,33)(21,37)(22,26)(23,31)(24,35)(25,29)(28,32)(30,41)(34,40)(36,42)(38,43)(39,44)(45,48)(46,47);(1,2)(3,6,11,9)(4,7)(5,8)(10,16,13,15)(12,14)(17,18)(19,28,31,43)(20,24,35,33)(21,25)(22,26)(23,32,27,38)(29,37)(30,46)(34,45)(36,42,39,44)(40,48)(41,47);(1,3)(2,6,8,9)(4,13)(5,11)(7,16,12,15)(10,14)(17,19,29,32,22,31,21,38)(18,23,37,28,26,27,25,43)(20,34,36,47,35,30,39,48)(24,41,42,46,33,40,44,45);(1,4,5,14)(2,7,8,12)(3,10,11,13)(6,15,9,16)(17,20,22,35)(18,24)(19,30,31,34)(21,39,29,36)(23,40)(25,44)(26,33)(27,41)(28,45)(32,48,38,47)(37,42)(43,46)

Polycyclic presentation:

2^10|b^2=g,c^2=efj,d^2=hi,f^2=j,b^a=bg,c^a=ce,c^b=cf,d^a=dgij,d^b=dgh,d^c=di,e^c=ei,e^d=ei,f^a=fi,f^b=fij,f^c=fi,f^d=fi,g^c=gi,g^d=gj,h^a=hj,h^b=hj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285435
maximal normal subgroup of	#72285435

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.5*F.6*F.10);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);