#6860273

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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

1237 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6767469);