#6557787

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1310 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=f,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6464983);