#6889915

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1341 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)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,30,36)(27,37,40,32)(28,42,43,34)(29,35,31,33)(38,47,52,51)(39,46,41,48)(44,50,45,49)(53,55,54,56);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,30,40)(26,32,36,37)(28,38,43,52)(29,39,31,41)(33,46,35,48)(34,47,42,51)(44,53,45,54)(49,55,50,56);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(26,33)(28,43)(29,31)(32,46)(34,50)(35,36)(37,48)(38,52)(39,41)(42,49)(47,56)(51,55);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(25,28,30,43)(26,34,36,42)(27,38,40,52)(29,45,31,44)(32,47,37,51)(33,49,35,50)(39,54,41,53)(46,55,48,56);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,29,30,31)(26,35,36,33)(27,39,40,41)(28,44,43,45)(32,48,37,46)(34,49,42,50)(38,53,52,54)(47,55,51,56)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6797111);