#7658782

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

4650 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=g,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, 7565978);