#7107557

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1240 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:

(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,37)(28,34)(29,35)(30,36)(31,33)(32,40)(38,50)(39,46)(41,48)(42,47)(43,49)(44,51)(45,52)(53,56)(54,55);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(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);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(26,33)(29,31)(32,46)(34,49)(35,36)(37,48)(39,41)(43,45)(47,55)(50,56)(51,52)(53,54);(3,8)(6,7)(12,20)(15,24)(17,19)(21,23)(25,28,30,44)(26,34,36,51)(27,38,40,42)(29,45,31,43)(32,47,37,50)(33,49,35,52)(39,54,41,53)(46,55,48,56);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(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=fhi,b^2=f,c^2=h,d^2=i,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, 7014753);