#7657906

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

4920 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,e^2=g,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, 7565102);