#6648814

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1231 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,d^2=hi,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,e^c=ei,g^c=gi,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, 6556010);