#8458588

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

4942 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)(3,8,4,7)(9,10,13,15)(11,12,16,14)(17,18)(19,23)(20,30)(21,26)(22,27)(24,28)(25,29)(31,32);(1,3,6,4)(2,7,5,8)(9,11,13,16)(10,14,15,12)(17,19,24,22)(18,23,28,27)(20,31,29,21)(25,26,30,32);(9,11)(10,12)(13,16)(14,15)(17,20)(18,25)(19,21)(22,31)(23,32)(24,29)(26,27)(28,30);(2,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,21,24,31)(18,26,28,32)(19,29,22,20)(23,25,27,30);(1,4,6,3)(2,8,5,7)(17,22,24,19)(18,27,28,23)(20,31,29,21)(25,26,30,32);(1,5,6,2)(3,8,4,7)(17,23,24,27)(18,19,28,22)(20,32,29,26)(21,30,31,25)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8365784);