#482312

order	512
nilpotency class	4
isoclinism	isoclinic class of #481619
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×C₄⋊C₄	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

818 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)(3,11,13,16)(4,9)(5,10)(6,8)(7,12,15,14)(17,18,22,27,23,26,37,42)(19,28,30,43,21,24,35,39)(20,31,33,44,34,25,36,41)(29,38,45,47,32,40,46,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,23,21)(18,24,26,28)(20,32,34,29)(22,30,37,35)(25,40,31,38)(27,43,42,39)(33,46,36,45)(41,47,44,48);(2,8)(7,15)(9,10)(11,16)(17,20,23,34)(18,25,26,31)(19,29,21,32)(22,36,37,33)(24,38,28,40)(27,44,42,41)(30,46,35,45)(39,48,43,47);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(17,21,23,19)(18,24,26,28)(20,29,34,32)(22,35,37,30)(25,40,31,38)(27,39,42,43)(33,45,36,46)(41,48,44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389508);