#6787897

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1177 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=h,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,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, 6695093);