#482168

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×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

843 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)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18)(19,29,20,25)(21,26)(22,27,36,44)(23,28)(24,31)(30,41,32,39)(33,38)(34,42,47,48)(35,43)(37,45)(40,46);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,23,20)(18,24,28,38)(21,32,35,30)(22,31,36,33)(25,27,29,44)(26,40,43,37)(34,46,47,45)(39,48,41,42);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,23,19)(18,25,28,29)(21,30,35,32)(22,33,36,31)(24,27,38,44)(26,41,43,39)(34,45,47,46)(37,48,40,42);(17,21,23,35)(18,26,28,43)(19,30,20,32)(22,34,36,47)(24,37,38,40)(25,39,29,41)(27,42,44,48)(31,45,33,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=i,e^2=h,b^a=be,c^a=cgh,e^a=eg,g^a=gi,d^b=di,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, 389364);