#8701700

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

5005 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)(9,10,13,15)(11,16,12,14)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,31)(29,32);(1,3,6,5)(2,4,7,8)(9,11,13,12)(10,14,15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,30)(31,32);(9,11,13,12)(10,14,15,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,31)(28,29)(30,32);(2,7)(3,5)(9,10)(11,14)(12,16)(13,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,29)(28,31)(30,32);(1,4,6,8)(2,3,7,5)(9,10)(11,14)(12,16)(13,15)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28);(1,5,6,3)(2,4,7,8)(9,12,13,11)(10,14,15,16)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 8608896);