#6831238

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1300 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,30,34)(27,31,37,43)(28,32,39,45)(29,33,40,46)(35,41,48,52)(36,42,49,53)(38,44,50,54)(47,51,55,56);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(27,35)(28,38)(29,40)(31,41)(32,44)(33,46)(36,49)(37,48)(39,50)(42,53)(43,52)(45,54);(2,5)(6,8)(25,27,29,35,30,37,40,48)(26,31,33,41,34,43,46,52)(28,36,38,47,39,49,50,55)(32,42,44,51,45,53,54,56);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(28,39)(32,45)(36,49)(38,50)(42,53)(44,54)(47,55)(51,56);(1,3,4,7)(2,6,5,8)(11,19)(14,17)(18,24)(22,23)(25,28,40,50,30,39,29,38)(26,32,46,54,34,45,33,44)(27,36,48,55,37,49,35,47)(31,42,52,56,43,53,41,51)

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=f,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6738434);