#483224

order	512
nilpotency class	4
isoclinism	isoclinic class of #482075
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	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

739 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390420);