#6654358

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632074
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1175 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6561554);