#8701733

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

5867 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238575, #72238577, #72238581
maximal normal subgroup of	#72238575, #72238577, #72238581

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8608929);