#8174644

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

4520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8081840);