#6644463

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632082
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1242 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=efhi,c^b=ch,d^b=dfh,e^b=ei,g^c=gi,e^d=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, 6551659);