#199819

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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1030 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)(3,5)(4,11)(6,9)(7,12)(8,10)(13,16)(14,15)(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);(1,3,10,9,6,8,5,2)(4,11,12,16,14,15,13,7)(17,19,22,29)(18,23)(20,27,31,44)(21,32,34,28)(24,35)(25,40)(26,37)(30,45,33,43)(36,42)(38,48)(39,47)(41,46);(1,4,6,14)(2,7,9,16)(3,11,8,15)(5,13,10,12)(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);(1,5,6,10)(2,3,9,8)(4,12,14,13)(7,15,16,11)(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=egh,c^2=h,d^2=h,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, 107015);