#7128260

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=h,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7035456);