#7129415

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1153 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fhi,d^2=h,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=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7036611);