#8422094

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

4353 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10,16,15)(11,20,19,14)(12,22,24,17)(13,21,23,18)(25,26,30,33)(27,31,35,29)(28,36,38,32)(34,37,40,39);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,21)(13,22)(15,20)(16,19)(17,23)(18,24)(25,27,30,35)(26,29,33,31)(28,34,38,40)(32,37,36,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,21)(13,20)(14,23)(15,22)(16,24)(18,19)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(1,3)(2,6)(4,8)(5,7)(9,13,16,23)(10,18,15,21)(11,17,19,22)(12,20,24,14)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,37,31,39);(9,14,16,20)(10,19,15,11)(12,23,24,13)(17,18,22,21);(1,2)(3,5)(4,7)(6,8)(9,15,16,10)(11,20,19,14)(12,22,24,17)(13,18,23,21)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,34,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,e^2=h,f^2=g,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, 8329290);