#483215

order	512
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

749 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)(17,18,22,27,20,25,35,26,23,28,38,43,36,44,21,37)(19,29,32,41,34,45,48,39,33,46,47,24,30,40,31,42);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,15)(10,16)(17,19)(18,24)(20,30)(21,32)(22,31)(23,33)(25,39)(26,45)(27,46)(28,41)(29,43)(34,36)(35,47)(37,40)(38,48)(42,44);(2,8)(7,11)(9,10)(15,16)(17,20,23,36)(18,25,28,44)(19,30,33,34)(21,22,35,38)(24,39,41,42)(26,43,37,27)(29,40,46,45)(31,47,48,32);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,15,11,16)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=gi,e^2=hi,f^2=ghi,h^2=i,b^a=be,c^a=cgi,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, 390411);