#6640531

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636940
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1248 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)(27,34)(28,39)(29,32)(30,35)(31,45)(33,41)(36,44)(37,51)(38,47)(40,54)(42,53)(43,50)(46,56)(48,55)(49,52);(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)(26,32)(28,37)(29,30)(31,38)(33,46)(34,35)(36,47)(39,53)(40,42)(41,52)(43,44)(45,50)(48,49)(51,54)(55,56);(3,7)(5,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28,31,41,30,40,44,55)(26,33,36,39,35,48,45,54)(27,37,38,52,29,42,43,56)(32,46,47,53,34,49,50,51);(3,7)(5,8)(10,17)(11,20)(16,22)(19,21)(25,29,30,27)(26,34,35,32)(28,37,40,42)(31,43,44,38)(33,46,48,49)(36,50,45,47)(39,53,54,51)(41,52,55,56);(2,6)(5,8)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(26,35)(27,29)(28,40)(36,45)(38,43)(41,55)(46,49)(51,53)

Polycyclic presentation:

2^9|a^2=fi,c^2=fgi,d^2=i,e^2=fi,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,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6547727);