#8097141

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

5728 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)(10,11)(12,16)(17,18,23,26)(19,27,30,24)(20,29,31,21)(22,25,32,28);(2,5)(3,7)(9,10,13,11)(12,15,16,14)(17,19,23,30)(18,24,26,27)(20,28,31,25)(21,22,29,32);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14);(1,3)(2,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,21)(19,28)(22,24)(23,31)(25,30)(26,29)(27,32);(17,21)(18,20)(19,22)(23,29)(24,28)(25,27)(26,31)(30,32);(17,22)(18,25)(19,29)(20,24)(21,30)(23,32)(26,28)(27,31)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^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, 8004337);