#7335186

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1401 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fgi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7242382);