#7298253

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1334 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26)(27,32)(28,34)(29,35)(30,33)(31,36)(37,45)(38,44)(39,47)(40,48)(41,50)(42,46)(43,49)(51,55)(52,54)(53,56);(2,5)(3,7)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,31,39)(26,32,36,47)(28,37,41,53)(29,42,30,38)(33,44,35,46)(34,45,50,56)(40,52,43,51)(48,54,49,55);(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)(29,30)(32,44)(34,48)(35,36)(38,42)(40,43)(45,54)(46,47)(49,50)(51,52)(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)(27,37)(29,43)(30,40)(31,41)(32,45)(33,48)(35,49)(36,50)(38,52)(39,53)(42,51)(44,54)(46,55)(47,56);(1,3,4,7)(2,6,5,8)(10,18)(15,19)(16,17)(23,24)(25,29,31,30)(26,35,36,33)(27,38,39,42)(28,40,41,43)(32,44,47,46)(34,48,50,49)(37,51,53,52)(45,55,56,54)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,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, 7205449);