#8456856

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6049 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(1,2,6,5)(3,4,7,8)(9,10)(11,13)(12,15)(14,16)(18,20)(22,24);(1,3,6,7)(2,4,5,8)(10,11)(14,15)(19,23)(22,24);(1,4)(2,3)(5,7)(6,8)(9,10,13,11)(12,14,16,15)(17,18,21,20)(19,22,23,24);(1,5,6,2)(3,4,7,8)(9,11,13,10)(12,15,16,14)(17,19,21,23)(18,22,20,24);(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,24,23,22);(1,4)(2,7)(3,5)(6,8)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=ghi,d^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312512, #72342317, #72342326
maximal normal subgroup of	#72312512, #72342317, #72342326

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8364052);