#7183043

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1292 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=hi,c^2=f,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7090239);