#6765218

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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

1328 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)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,26,31,35)(27,32,30,45)(28,38,41,54)(29,34,43,50)(33,36,48,40)(37,46,44,56)(39,53,42,55)(47,51,49,52);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24)(25,27,31,30)(26,32,35,45)(28,36,41,40)(29,37,43,44)(33,38,48,54)(34,46,50,56)(39,51,42,52)(47,53,49,55);(2,6)(3,7)(25,28,30,40,31,41,27,36)(26,33,32,38,35,48,45,54)(29,42,44,51,43,39,37,52)(34,49,46,55,50,47,56,53);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,29)(26,34)(27,37)(28,39)(30,44)(31,43)(32,46)(33,47)(35,50)(36,51)(38,53)(40,52)(41,42)(45,56)(48,49)(54,55);(3,7)(5,8)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(29,43)(34,50)(37,44)(39,42)(46,56)(47,49)(51,52)(53,55)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6672414);