#6831270

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

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^c=ei,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, 6738466);