#8422090

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

4772 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8329286);