#7128269

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=h,d^2=hi,e^2=f,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, 7035465);