#8196991

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

5406 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)(19,25)(20,28)(21,23)(22,27)(24,29)(26,30)(31,32);(2,3)(6,7)(9,10,13,12)(11,15,16,14)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32);(4,8)(6,7)(10,12)(14,15)(18,23)(22,25)(24,28)(30,31);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,14,12,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32);(17,20,21,29)(18,24,23,28)(19,26,27,32)(22,30,25,31);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,15,16,14)(17,18)(19,22)(20,24)(21,23)(25,27)(26,30)(28,29)(31,32)

Polycyclic presentation:

2^9|b^2=i,d^2=ghi,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, 8104187);