#6464863

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

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,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, 6372059);