#6582892

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1344 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,29,33,24)(20,34,37,25)(21,26,32,45)(23,28,31,43)(30,40,36,47)(35,41,48,44)(38,46,39,42);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,35)(21,23)(22,33)(25,41)(26,43)(27,29)(28,45)(30,38)(31,32)(34,44)(36,39)(37,48)(40,42)(46,47);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,20,22,37)(18,25,27,34)(19,30,33,36)(21,35,32,48)(23,39,31,38)(24,40,29,47)(26,44,45,41)(28,46,43,42);(1,3,5,4)(2,7,8,6)(17,21)(18,26)(19,31)(20,36)(22,32)(23,33)(24,28)(25,40)(27,45)(29,43)(30,37)(34,47)(35,38)(39,48)(41,46)(42,44);(1,4,5,3)(2,6,8,7)(19,32)(21,33)(23,31)(24,26)(28,43)(29,45)(30,48)(35,36)(38,39)(40,44)(41,47)(42,46)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490088);