#7187292

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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

1191 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=fg,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7094488);