#483282

order	512
nilpotency class	4
isoclinism	isoclinic class of #482956
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	#62474	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

693 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=i,e^2=hi,f^2=ghi,h^2=i,b^a=be,e^a=eg,e^b=eh,f^b=fgh,h^b=hi,d^c=dg

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390478);