#8197335

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

4980 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,20)(21,30)(22,25)(23,26)(24,27)(28,31)(29,32);(1,2)(3,5)(4,6)(7,8)(10,13)(14,15)(17,19)(18,21)(20,27)(22,28)(23,29)(24,30)(25,26)(31,32);(1,2,4,6)(3,5,8,7)(9,10)(11,14)(12,13)(15,16)(17,20,24,21)(18,19,27,30)(22,31,29,26)(23,25,28,32);(9,11,12,16)(10,14,13,15)(17,21,24,20)(18,19,27,30)(22,26,29,31)(23,25,28,32);(1,3)(2,5)(4,8)(6,7)(17,22,24,29)(18,25,27,32)(19,28,30,23)(20,31,21,26);(1,2)(3,5)(4,6)(7,8)(17,23,24,28)(18,26,27,31)(19,29,30,22)(20,32,21,25)

Polycyclic presentation:

2^9|c^2=gi,d^2=g,e^2=ghi,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, 8104531);