#8701726

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
prime radical	#36412	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4781 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,6,7)(3,8,5,4)(9,10,12,13)(11,15,16,14)(17,18,23,21)(19,24,26,20)(22,25,31,29)(27,30,32,28);(1,3,6,5)(2,4,7,8)(10,13)(14,15)(19,26)(20,24)(27,32)(28,30);(9,10)(11,14)(12,13)(15,16)(17,19,23,26)(18,24,21,20)(22,27,31,32)(25,30,29,28);(2,7)(3,5)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,26)(22,30)(23,24)(25,32)(27,29)(28,31);(1,4,6,8)(2,3,7,5)(9,11,12,16)(10,14,13,15)(17,21,23,18)(19,20,26,24)(22,25,31,29)(27,30,32,28);(1,5,6,3)(2,4,7,8)(17,22,23,31)(18,25,21,29)(19,27,26,32)(20,28,24,30)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,e^2=ghi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8608922);