#6589182

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1313 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=fgi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,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, 6496378);