#6433961

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433057
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1590 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341157);