#7658983

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

5186 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)(17,18,23,24)(19,27,21,25)(20,22,26,32)(28,31,30,29);(1,2)(3,5)(4,6)(7,8)(10,13)(14,16)(18,24)(20,26)(25,27)(28,30);(9,10)(11,14)(12,13)(15,16)(17,19,23,21)(18,25,24,27)(20,28,26,30)(22,29,32,31);(9,11,12,15)(10,14,13,16)(17,20,23,26)(18,22,24,32)(19,28,21,30)(25,29,27,31);(1,3,4,7)(2,5,6,8)(17,21,23,19)(18,25,24,27)(20,30,26,28)(22,29,32,31);(17,22)(18,26)(19,29)(20,24)(21,31)(23,32)(25,30)(27,28)

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=h,e^2=g,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, 7566179);