#6826401

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=hi,d^2=g,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6733597);