#8425059

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

4907 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=ghi,e^2=h,f^2=h,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, 8332255);