#7658229

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

4862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|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, 7565425);