#482897

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	C₂×#120	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#551 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,d^2=i,e^2=hi,b^a=be,e^a=eg,g^a=gi,c^b=ci,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, 390093);