#6859935

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1234 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)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,35)(27,32,37,47)(28,39,42,51)(29,34,30,44)(33,45,52,41)(36,53,40,56)(38,48,43,49)(46,54,50,55);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,32)(28,40)(29,43)(30,38)(31,37)(33,50)(34,49)(35,47)(36,42)(39,56)(41,54)(44,48)(45,55)(46,52)(51,53);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,31,42)(26,33,35,52)(27,36,37,40)(29,41,30,45)(32,46,47,50)(34,51,44,39)(38,54,43,55)(48,53,49,56);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(27,37)(32,47)(36,40)(38,43)(46,50)(48,49)(53,56)(54,55);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(25,29,31,30)(26,34,35,44)(27,38,37,43)(28,41,42,45)(32,48,47,49)(33,51,52,39)(36,54,40,55)(46,53,50,56)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6767131);