#482612

order	512
nilpotency class	4
isoclinism	isoclinic class of #482397
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

827 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)(19,29,21,26)(20,25)(22,27,36,44)(23,28)(24,31)(30,40,32,43)(33,41,47,48)(34,42)(35,38)(37,45)(39,46);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,23,21)(18,24,28,38)(20,32,34,30)(22,31,36,35)(25,39,42,37)(26,27,29,44)(33,46,47,45)(40,41,43,48);(2,8)(7,15)(9,10)(11,16)(17,20,23,34)(18,25,28,42)(19,30,21,32)(22,33,36,47)(24,37,38,39)(26,43,29,40)(27,41,44,48)(31,45,35,46);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(17,21,23,19)(18,26,28,29)(20,30,34,32)(22,35,36,31)(24,27,38,44)(25,40,42,43)(33,45,47,46)(37,48,39,41)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=hi,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,d^c=di,f^d=fi,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, 389808);