#199821

order	512
nilpotency class	3
isoclinism	isoclinic class of #199815
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

979 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)(13,15)(17,18)(19,23,29,37)(20,24)(21,25)(22,26)(27,35,44,47)(28,36,32,40)(30,38)(31,39)(33,41)(34,42)(43,46,45,48);(1,2,4,6,5,8,12,14)(3,7,11,13,10,16,9,15)(17,19)(18,23,26,37)(20,27)(21,32)(22,29)(24,35,39,47)(25,40,42,36)(28,34)(30,45)(31,44)(33,43)(38,48,41,46);(1,3,5,10)(2,7,8,16)(4,11,12,9)(6,13,14,15)(17,20,22,31)(18,24,26,39)(19,27,29,44)(21,33,34,30)(23,35,37,47)(25,41,42,38)(28,45,32,43)(36,48,40,46);(2,8)(3,10)(6,14)(9,11)(17,21,22,34)(18,25,26,42)(19,28,29,32)(20,30,31,33)(23,36,37,40)(24,38,39,41)(27,43,44,45)(35,46,47,48)

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,d^b=dh,e^b=ei,g^b=gi,d^c=dh,f^c=fhi

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, 107017);