#8533359

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

4179 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440555);