#6891201

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei,g^c=gi,e^d=ei

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, 6798397);