#7298388

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234345
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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1311 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7205584);