#483230

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	#60718	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

722 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)(17,18,20,25,32,40,44,36,22,27,33,35,21,26,31,23)(19,24,29,38,42,47,48,45,28,37,41,46,30,39,43,34);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,15)(10,16)(18,23)(20,31)(21,32)(24,34)(25,26)(27,36)(29,43)(30,42)(33,44)(35,40)(37,45)(38,39)(41,48)(46,47);(2,8)(7,11)(9,10)(15,16)(17,19,22,28)(18,24,27,37)(20,29,33,41)(21,30,32,42)(23,34,36,45)(25,38,35,46)(26,39,40,47)(31,43,44,48);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,15,11,16)(19,28)(24,37)(29,41)(30,42)(34,45)(38,46)(39,47)(43,48)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=g,e^2=hi,f^2=ghi,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, 390426);