#7129673

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fhi,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=eh,d^c=dhi,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, 7036869);