#8679873

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

4656 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8587069);