#72308709

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#11022
commutator	C₂×C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₄×C₈	C₂³
prime radical	#7623354	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1746 ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=f,c^2=gi,d^2=hj,g^2=ij,i^2=j,b^a=be,c^a=cg,c^b=cd,d^a=dh,d^b=dj,e^c=ej,f^c=fh,g^a=gi,g^b=ghj,i^a=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308734, #72308826
maximal normal subgroup of	#72308734, #72308826

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