#6486601

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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

1178 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6393797);