#7656896

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

4725 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:

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

Polycyclic presentation:

2^9|b^2=h,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, 7564092);