#6654473

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
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)(25,26,30,34)(27,31,36,45)(28,33,40,50)(29,32,42,35)(37,44,39,48)(38,46,51,43)(41,47,54,49)(52,56,53,55);(1,2)(3,6)(4,5)(7,8)(9,11,14,20)(10,15,17,18)(12,19,13,21)(16,23,22,24)(25,27,30,36)(26,31,34,45)(28,39,40,37)(29,38,42,51)(32,46,35,43)(33,48,50,44)(41,53,54,52)(47,55,49,56);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(26,32)(27,36)(29,42)(31,43)(33,47)(34,35)(37,39)(41,54)(44,55)(45,46)(48,56)(49,50);(2,5)(6,8)(10,17)(11,20)(16,22)(19,21)(25,28)(26,33)(27,37)(29,41)(30,40)(31,44)(32,47)(34,50)(35,49)(36,39)(38,52)(42,54)(43,55)(45,48)(46,56)(51,53);(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)(27,36)(28,40)(32,35)(38,51)(41,54)(44,48)(55,56)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,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, 6561669);