#483078

order	512
nilpotency class	4
isoclinism	isoclinic class of #481466
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

721 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:

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

Polycyclic presentation:

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

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, 390274);