#8508881

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

4365 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=hi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8416077);