#6640422

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636852
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1227 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,20,15)(12,16,13,22)(19,24,21,23)(25,26)(27,32)(28,40)(29,34)(30,44)(31,35)(33,41)(36,50)(37,46)(38,47)(39,54)(42,48)(43,49)(45,51)(52,55)(53,56);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24)(25,27,31,38)(26,32,35,47)(28,36,42,52)(29,37,30,39)(33,45,49,56)(34,46,44,54)(40,50,48,55)(41,51,43,53);(3,7)(5,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28,30,41,31,42,29,43)(26,33,34,40,35,49,44,48)(27,36,39,51,38,52,37,53)(32,45,46,50,47,56,54,55);(3,7)(5,8)(10,17)(11,20)(16,22)(19,21)(25,29,31,30)(26,34,35,44)(27,37,38,39)(28,41,42,43)(32,46,47,54)(33,48,49,40)(36,51,52,53)(45,55,56,50);(2,6)(5,8)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(27,38)(32,47)(36,52)(37,39)(45,56)(46,54)(50,55)(51,53)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6547618);