#8271129

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

4941 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:

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

Polycyclic presentation:

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

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, 8178325);