#6709666

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1229 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26,31,36)(27,32,40,48)(28,33,43,51)(29,34,30,35)(37,45,54,56)(38,46,39,47)(41,49,53,55)(42,50,44,52);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,31,40)(26,32,36,48)(28,41,43,53)(29,38,30,39)(33,49,51,55)(34,46,35,47)(37,44,54,42)(45,52,56,50);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,31,43)(26,33,36,51)(27,37,40,54)(29,42,30,44)(32,45,48,56)(34,50,35,52)(38,53,39,41)(46,55,47,49);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,29,31,30)(26,34,36,35)(27,38,40,39)(28,42,43,44)(32,46,48,47)(33,50,51,52)(37,53,54,41)(45,55,56,49);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(27,39)(29,30)(32,47)(34,35)(37,41)(38,40)(42,44)(45,49)(46,48)(50,52)(53,54)(55,56)

Polycyclic presentation:

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

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, 6616862);