#8455725

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

4731 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:

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

Polycyclic presentation:

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8362921);