#7658251

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

4640 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)(10,13)(11,15)(17,18)(19,26)(20,21)(22,25)(23,24)(27,32)(28,30)(29,31);(1,2)(3,7)(4,8)(5,6)(10,13)(14,16)(18,24)(20,29)(25,26)(27,30);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,22)(18,25,24,26)(20,27,29,30)(21,28,31,32);(9,11)(10,14)(12,15)(13,16)(17,20)(18,21)(19,27)(22,30)(23,29)(24,31)(25,28)(26,32);(1,4,5,3)(2,8,6,7)(17,21)(18,20)(19,28)(22,32)(23,31)(24,29)(25,27)(26,30);(17,22,23,19)(18,25,24,26)(20,30,29,27)(21,28,31,32)

Polycyclic presentation:

2^9|c^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, 7565447);