#6763673

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

1349 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)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26,30,35)(27,34,38,31)(28,39,42,48)(29,43,46,33)(32,40,50,36)(37,47,45,52)(41,56,44,55)(49,53,51,54);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,38)(26,31,35,34)(28,40,42,36)(29,37,46,45)(32,48,50,39)(33,47,43,52)(41,54,44,53)(49,56,51,55);(2,6)(3,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,27,36,30,42,38,40)(26,32,31,39,35,50,34,48)(29,44,37,54,46,41,45,53)(33,51,47,56,43,49,52,55);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29,30,46)(26,33,35,43)(27,37,38,45)(28,41,42,44)(31,47,34,52)(32,49,50,51)(36,53,40,54)(39,55,48,56);(3,7)(5,8)(11,21)(15,18)(19,20)(23,24)(25,27,30,38)(26,34,35,31)(28,36,42,40)(29,45,46,37)(32,48,50,39)(33,47,43,52)(41,54,44,53)(49,55,51,56)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgh,d^2=f,e^2=i,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, 6670869);