#7658366

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

4636 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7565562);