#7658882

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

4564 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=g,e^2=gi,f^2=gi,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, 7566078);