#9210381

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

4509 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=i,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9117577);