#7658872

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

4938 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=g,e^2=g,f^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, 7566068);