#7657895

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

4737 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)(11,14)(12,16)(13,15)(17,18)(19,23)(20,30)(21,26)(22,27)(24,28)(25,29)(31,32);(1,3,5,8)(2,6,4,7)(9,11,13,16)(10,14,15,12)(17,19,24,21)(18,23,28,26)(20,22,29,31)(25,32,30,27);(1,2,5,4)(3,6,8,7)(9,11)(10,12)(13,16)(14,15)(17,20)(18,25)(19,22)(21,31)(23,32)(24,29)(26,27)(28,30);(9,12)(10,11)(13,14)(15,16)(17,21,24,19)(18,26,28,23)(20,31,29,22)(25,27,30,32);(1,4,5,2)(3,7,8,6)(17,22,24,31)(18,27,28,32)(19,29,21,20)(23,25,26,30);(17,23,24,26)(18,19,28,21)(20,32,29,27)(22,30,31,25)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=hi,e^2=ghi,f^2=hi,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, 7565091);