#8422193

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

4788 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:

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

Polycyclic presentation:

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

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, 8329389);