#6589598

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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1594 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,d^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,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, 6496794);