#7335435

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fgi,d^2=i,e^2=fi,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=dh,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, 7242631);