#483203

order	512
nilpotency class	4
isoclinism	isoclinic class of #482080
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#60717	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=g,e^2=hi,f^2=gh,h^2=i,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,h^b=hi,d^c=di

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, 390399);