#7658223

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

5508 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|e^2=gh,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, 7565419);