#8992935

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

4628 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8900131);