#7656807

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

4795 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)(10,13)(11,15)(17,18,23,22)(19,27,30,24)(20,31,29,25)(21,26,28,32);(1,2)(3,6)(4,7)(5,8)(10,13)(14,16)(17,19)(18,24)(20,28)(21,29)(22,27)(23,30)(25,32)(26,31);(9,10)(11,14)(12,13)(15,16)(17,20,23,29)(18,25,22,31)(19,28,30,21)(24,32,27,26);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,21,23,28)(18,26,22,32)(19,29,30,20)(24,31,27,25);(1,4,5,3)(2,7,8,6)(17,19)(18,27)(20,28)(21,29)(22,24)(23,30)(25,26)(31,32);(17,22,23,18)(19,27,30,24)(20,31,29,25)(21,32,28,26)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7564003);