#6915998

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907977
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

1350 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6823194);