#6587615

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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 ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1323 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,e^c=ei,h^c=hi,e^d=ehi,h^d=hi,h^e=hi

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, 6494811);