#6756125

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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

1348 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)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,26,31,36)(27,32,38,46)(28,41,43,50)(29,35,30,34)(33,44,49,42)(37,51,52,56)(39,48,40,47)(45,54,55,53);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24)(25,27,31,38)(26,32,36,46)(28,37,43,52)(29,39,30,40)(33,45,49,55)(34,47,35,48)(41,51,50,56)(42,53,44,54);(1,3,4,8)(2,5,6,7)(25,28)(26,33)(27,37)(29,42)(30,44)(31,43)(32,45)(34,41)(35,50)(36,49)(38,52)(39,53)(40,54)(46,55)(47,51)(48,56);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(27,38)(32,46)(37,52)(39,40)(45,55)(47,48)(51,56)(53,54);(2,6)(5,7)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,29,31,30)(26,34,36,35)(27,39,38,40)(28,42,43,44)(32,47,46,48)(33,41,49,50)(37,53,52,54)(45,51,55,56)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6663321);