#6889952

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885065
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

1330 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)(25,26,30,35)(27,37,40,32)(28,41,43,48)(29,36,31,34)(33,42,49,44)(38,52,54,56)(39,47,45,50)(46,53,51,55);(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)(26,32,35,37)(28,38,43,54)(29,45,31,39)(33,46,49,51)(34,50,36,47)(41,56,48,52)(42,55,44,53);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,28,30,43)(26,33,35,49)(27,38,40,54)(29,44,31,42)(32,46,37,51)(34,41,36,48)(39,55,45,53)(47,52,50,56);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(25,29,30,31)(26,34,35,36)(27,39,40,45)(28,42,43,44)(32,47,37,50)(33,48,49,41)(38,53,54,55)(46,56,51,52);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,40)(32,37)(38,54)(39,45)(46,51)(47,50)(52,56)(53,55)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6797148);