#6587253

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

1325 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)(19,28)(20,25)(21,24)(22,26)(23,27)(29,47)(30,39)(31,45)(32,41)(33,42)(34,43)(35,38)(36,44)(37,46)(40,48);(1,3,5,4)(2,6,7,8)(17,19)(18,24)(20,32)(21,22)(23,30)(25,40)(26,28)(27,45)(29,34)(31,36)(33,48)(35,37)(38,43)(39,44)(41,42)(46,47);(10,13)(14,16)(17,20,22,33)(18,25,26,42)(19,29,21,35)(23,37,36,34)(24,38,28,47)(27,46,44,43)(30,32,31,48)(39,41,45,40);(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,22)(20,32)(23,31)(25,41)(26,28)(27,45)(29,34)(30,36)(33,48)(35,37)(38,46)(39,44)(40,42)(43,47);(1,4,5,3)(2,6,7,8)(18,26)(19,30)(21,31)(23,36)(24,39)(25,42)(28,45)(29,32)(34,37)(35,48)(38,41)(40,47)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=g,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=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, 6494449);