#6460643

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1617 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)(11,13)(12,14)(15,16)(17,18)(19,29)(20,32)(21,25)(22,27)(23,28)(24,38)(26,36)(30,44)(31,41)(33,46)(34,43)(35,47)(37,40)(39,48)(42,45);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,33)(21,23)(22,37)(24,31)(26,42)(27,41)(28,29)(30,35)(32,44)(34,45)(36,48)(38,40)(39,43)(46,47);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20,24,36,23,35,22,34)(18,26,27,32,28,43,38,47)(19,30,31,45,21,33,37,48)(25,39,41,46,29,42,40,44);(1,4,5,3)(2,6,8,7)(11,15)(13,16)(17,21,23,19)(18,25,28,29)(20,33,35,30)(22,31,24,37)(26,39,43,42)(27,41,38,40)(32,46,47,44)(34,45,36,48);(17,22,23,24)(18,27,28,38)(19,31,21,37)(20,34,35,36)(25,40,29,41)(26,32,43,47)(30,45,33,48)(39,44,42,46)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fgh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^d=ei

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, 6367839);