#6889854

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1323 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)(25,26,31,36)(27,32,41,49)(28,34,43,51)(29,35,30,45)(33,44,52,42)(37,53,38,47)(39,48,40,56)(46,55,50,54);(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)(26,32)(28,38)(29,39)(30,40)(31,41)(33,50)(34,47)(35,48)(36,49)(37,43)(42,55)(44,54)(45,56)(46,52)(51,53);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,28,31,43)(26,33,36,52)(27,37,41,38)(29,42,30,44)(32,46,49,50)(34,35,51,45)(39,54,40,55)(47,56,53,48);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(25,28)(26,34)(27,38)(29,44)(30,42)(31,43)(32,47)(33,45)(35,52)(36,51)(37,41)(39,54)(40,55)(46,48)(49,53)(50,56);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,29,31,30)(26,35,36,45)(27,39,41,40)(28,42,43,44)(32,48,49,56)(33,51,52,34)(37,54,38,55)(46,47,50,53)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,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, 6797050);