#6464873

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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

1381 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)(17,18,22,28,23,27,38,44)(19,29,32,42,33,46,39,24)(20,34,35,45,36,25,21,26)(30,41,31,40,37,43,47,48);(2,5)(6,8)(17,19,23,33)(18,24,27,42)(20,31,36,47)(21,37,35,30)(22,39,38,32)(25,43,34,41)(26,40,45,48)(28,46,44,29);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,22,35,23,36,38,21)(18,25,28,26,27,34,44,45)(19,30,32,31,33,37,39,47)(24,40,29,43,42,48,46,41);(17,21,38,36,23,35,22,20)(18,26,44,25,27,45,28,34)(19,31,39,30,33,47,32,37)(24,41,46,48,42,43,29,40);(9,11,12,15)(10,14,13,16)(18,27)(24,42)(25,34)(26,45)(28,44)(29,46)(40,48)(41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372069);