#6765839

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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

1365 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,31,35)(27,32,30,44)(28,38,41,51)(29,34,42,48)(33,36,47,40)(37,45,43,54)(39,50,52,56)(46,49,55,53);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,31,30)(26,32,35,44)(28,36,41,40)(29,37,42,43)(33,38,47,51)(34,45,48,54)(39,49,52,53)(46,50,55,56);(2,6)(3,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,30,40,31,41,27,36)(26,33,32,38,35,47,44,51)(29,39,43,53,42,52,37,49)(34,46,45,50,48,55,54,56);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,29)(26,34)(27,37)(28,39)(30,43)(31,42)(32,45)(33,46)(35,48)(36,49)(38,50)(40,53)(41,52)(44,54)(47,55)(51,56);(3,7)(5,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(29,42)(34,48)(37,43)(39,52)(45,54)(46,55)(49,53)(50,56)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgh,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,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, 6673035);