#482355

order	512
nilpotency class	4
isoclinism	isoclinic class of #482080
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60707	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

742 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)(3,9,11,6)(4,8)(5,7)(10,15,16,13)(12,14)(17,18,21,27,23,29,22,28)(19,25,32,42,30,40,33,43)(20,34,36,26,37,24,35,39)(31,45,47,41,44,38,46,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(18,24)(20,35)(21,22)(25,38)(26,27)(28,39)(29,34)(31,46)(32,33)(36,37)(40,45)(41,42)(43,48)(44,47);(2,7)(6,9)(8,14)(13,15)(17,19)(18,25)(20,31)(21,32)(22,33)(23,30)(24,38)(26,41)(27,42)(28,43)(29,40)(34,45)(35,46)(36,47)(37,44)(39,48);(1,4,5,12)(2,8,7,14)(3,10,11,16)(6,13,9,15)(19,30)(25,40)(31,44)(32,33)(38,45)(41,48)(42,43)(46,47)

Polycyclic presentation:

2^9|a^2=f,d^2=gh,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cgh,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,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, 389551);