#6914870

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1375 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,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, 6822066);