#6589279

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=fgi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6496475);