#6464943

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1378 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)(11,14)(12,13)(15,16)(17,18,22,28,24,30,23,29)(19,31,34,43,36,42,35,25)(20,26,21,27,37,44,38,45)(32,47,48,46,39,41,33,40);(2,5)(6,8)(17,19,24,36)(18,25,30,43)(20,33,37,48)(21,39,38,32)(22,35,23,34)(26,41,44,47)(27,46,45,40)(28,42,29,31);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,22,21,24,37,23,38)(18,26,28,27,30,44,29,45)(19,32,34,48,36,39,35,33)(25,40,31,47,43,46,42,41);(17,21,22,37,24,38,23,20)(18,27,28,44,30,45,29,26)(19,33,34,32,36,48,35,39)(25,41,31,40,43,47,42,46);(9,11,12,15)(10,14,13,16)(20,37)(21,38)(26,44)(27,45)(32,39)(33,48)(40,46)(41,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=hi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=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, 6372139);