#72312705

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1439
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	C₂²×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

3917 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=ij,b^2=h,c^2=fij,d^2=ij,e^2=j,f^2=j,c^a=cij,c^b=cf,d^b=de,d^c=deg,e^b=ei,e^c=ej,e^d=ej,f^b=fj,f^d=fi,g^b=gj,g^c=gi,h^d=hij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312723, #72312912, #72313478
maximal normal subgroup of	#72312723, #72312912, #72313478

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, 1, F.9*F.10);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.6*F.9*F.10);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);