#6856561

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1189 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6763757);