#482538

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×C₄⋊C₄	C₂⁴
prime radical	#36011	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

858 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=i,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,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, 389734);