#8422230

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

5377 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=g,e^2=gh,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, 8329426);