#8096240

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

4887 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)(11,15)(12,14)(13,16)(17,18)(19,27)(20,29)(21,23)(22,25)(24,26)(28,31)(30,32);(1,2)(3,6)(4,7)(5,8)(9,10,12,14)(11,13,16,15)(17,19,24,23)(18,21,26,27)(20,30,22,28)(25,32,29,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,20,24,22)(18,25,26,29)(19,28,23,30)(21,31,27,32);(9,11)(10,13)(12,16)(14,15)(17,21)(18,19)(20,31)(22,32)(23,26)(24,27)(25,28)(29,30);(1,3)(2,6)(4,8)(5,7)(17,22,24,20)(18,25,26,29)(19,28,23,30)(21,32,27,31);(1,2)(3,5)(4,7)(6,8)(17,23,24,19)(18,21,26,27)(20,30,22,28)(25,31,29,32)

Polycyclic presentation:

2^9|b^2=g,c^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003436);