#6640389

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,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,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, 6547585);