#199812

order	512
nilpotency class	3
isoclinism	isoclinic class of #199806
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

982 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=i,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,g^b=gi,d^c=dh,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107008);