#8110577

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

4768 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017773);