#6640504

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636908
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1252 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)(25,26,30,34)(27,31,36,44)(28,38,40,53)(29,42,39,33)(32,46,47,41)(35,48,50,56)(37,52,49,45)(43,54,55,51);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24)(25,27,30,36)(26,31,34,44)(28,35,40,50)(29,37,39,49)(32,43,47,55)(33,45,42,52)(38,48,53,56)(41,51,46,54);(3,7)(5,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28,39,46,30,40,29,41)(26,32,42,53,34,47,33,38)(27,35,49,54,36,50,37,51)(31,43,52,56,44,55,45,48);(3,7)(5,8)(10,17)(11,20)(16,22)(19,21)(28,40)(32,47)(35,50)(38,53)(41,46)(43,55)(48,56)(51,54);(2,6)(5,8)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(27,36)(28,40)(31,44)(32,47)(37,49)(38,53)(41,46)(45,52)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fgi,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,e^c=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, 6547700);