#6799416

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1226 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)(25,26,31,36)(27,32,37,48)(28,34,42,52)(29,35,30,33)(38,46,40,50)(39,47,43,45)(41,49,44,51)(53,55,54,56);(1,2,4,5)(3,7,8,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,31,37)(26,32,36,48)(28,40,42,38)(29,43,30,39)(33,47,35,45)(34,50,52,46)(41,53,44,54)(49,55,51,56);(2,5)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,33)(27,37)(29,30)(32,45)(34,49)(35,36)(38,40)(41,44)(46,55)(47,48)(50,56)(51,52);(10,16)(14,17)(15,22)(21,23)(25,28)(26,34)(27,38)(29,44)(30,41)(31,42)(32,46)(33,49)(35,51)(36,52)(37,40)(39,54)(43,53)(45,55)(47,56)(48,50);(1,3,4,8)(2,6,5,7)(11,19)(14,17)(18,24)(21,23)(25,29,31,30)(26,35,36,33)(27,39,37,43)(28,41,42,44)(32,47,48,45)(34,49,52,51)(38,53,40,54)(46,55,50,56)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,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,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6706612);