#7335173

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1379 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)(25,26,31,35)(27,36,37,47)(28,33,40,48)(29,34,30,44)(32,42,45,39)(38,51,52,56)(41,49,43,50)(46,53,55,54);(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)(32,45)(33,48)(36,47)(39,42)(41,43)(49,50);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,27,30,39,31,37,29,42)(26,32,34,36,35,45,44,47)(28,38,41,54,40,52,43,53)(33,46,50,51,48,55,49,56);(2,6)(3,8)(11,20)(15,18)(19,21)(23,24)(25,28)(26,33)(27,38)(29,43)(30,41)(31,40)(32,46)(34,50)(35,48)(36,51)(37,52)(39,54)(42,53)(44,49)(45,55)(47,56);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(25,29,31,30)(26,34,35,44)(27,39,37,42)(28,41,40,43)(32,47,45,36)(33,49,48,50)(38,53,52,54)(46,51,55,56)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fg,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=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, 7242369);