#6826399

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1130 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=h,d^2=gi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6733595);