#8657165

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

5307 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)(3,8)(4,6)(5,7)(17,18,23,27)(19,24,30,29)(20,31,21,25)(22,26,32,28);(2,4)(7,8)(9,10,15,16)(11,13,14,12)(17,19,23,30)(18,24,27,29)(20,28,21,26)(22,25,32,31);(1,3,6,5)(2,7,4,8)(9,11,15,14)(10,12,16,13)(17,20,23,21)(18,25,27,31)(19,26,30,28)(22,29,32,24);(1,2,6,4)(3,7,5,8)(9,12,15,13)(10,11,16,14)(17,21,23,20)(18,25,27,31)(19,28,30,26)(22,29,32,24);(1,4,6,2)(3,7,5,8)(9,13,15,12)(10,11,16,14)(17,18)(19,29)(20,31)(21,25)(22,26)(23,27)(24,30)(28,32);(1,5,6,3)(2,7,4,8)(9,14,15,11)(10,12,16,13)(17,22,23,32)(18,26,27,28)(19,25,30,31)(20,29,21,24)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=gi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8564361);