#8195869

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

5386 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)(3,5)(4,6)(7,8)(17,18,24,23)(19,27,22,25)(20,30,26,21)(28,29,31,32);(2,3)(6,7)(9,10,13,12)(11,15,16,14)(17,19,24,22)(18,25,23,27)(20,31,26,28)(21,32,30,29);(4,8)(6,7)(10,12)(14,15)(17,20)(18,21)(19,28)(22,31)(23,30)(24,26)(25,29)(27,32);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,21,24,30)(18,26,23,20)(19,29,22,32)(25,31,27,28);(17,22,24,19)(18,27,23,25)(20,31,26,28)(21,32,30,29);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,15,16,14)(17,23,24,18)(19,27,22,25)(20,30,26,21)(28,32,31,29)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=g,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8103065);