#8254898

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

4247 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:

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

Polycyclic presentation:

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

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, 8162094);