#6819993

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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1151 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)(4,8)(5,7)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,37)(28,38)(29,36)(30,34)(31,35)(32,39)(33,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,31,39)(26,32,35,37)(28,36,34,33)(29,38,40,30);(1,4,5,3)(2,6,7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,29,37,31,34,40,32)(26,30,36,39,35,38,33,27);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,29,31,40)(26,33,35,36)(27,38,39,30)(28,37,34,32);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,30,40,27,31,38,29,39)(26,34,33,37,35,28,36,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fgh,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6727189);