#8323211

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

4771 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8230407);