#8097739

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

5721 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=gh,e^2=ghi,f^2=ghi,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, 8004935);