#482859

order	512
nilpotency class	4
isoclinism	isoclinic class of #482728
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#552 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

774 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=i,d^2=i,e^2=hi,b^a=be,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fghi,d^c=dhi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390055);