#482311

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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

851 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)(17,18,21,26,22,25,36,42)(19,27,29,43,30,23,37,39)(20,31,34,44,33,24,35,41)(28,38,45,47,32,40,46,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,22,30)(18,23,25,27)(20,32,33,28)(21,29,36,37)(24,40,31,38)(26,43,42,39)(34,46,35,45)(41,47,44,48);(2,8)(7,15)(9,10)(11,16)(17,20,22,33)(18,24,25,31)(19,28,30,32)(21,35,36,34)(23,38,27,40)(26,44,42,41)(29,46,37,45)(39,48,43,47);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(20,33)(24,31)(28,32)(34,35)(38,40)(41,44)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,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, 389507);