#8096910

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

5068 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=g,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, 8004106);