#200102

order	512
nilpotency class	3
isoclinism	isoclinic class of #199842
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	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1052 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:

(1,2,5,9,6,10,4,7)(3,11,14,8,12,16,13,15)(17,18)(19,23,29,37)(20,24)(21,32)(22,26)(25,34)(27,35,42,47)(28,43,45,36)(30,40)(31,39)(33,38)(41,48,44,46);(2,7,10,9)(8,11,15,16)(17,19)(18,23,26,37)(20,27)(21,28)(22,29)(24,35,39,47)(25,36,32,43)(30,41)(31,42)(33,44)(34,45)(38,46,40,48);(1,3,6,12)(2,8,10,15)(4,13,5,14)(7,11,9,16)(17,20,22,31)(18,24,26,39)(19,27,29,42)(21,33,34,30)(23,35,37,47)(25,40,32,38)(28,44,45,41)(36,48,43,46);(3,12)(8,15)(11,16)(13,14)(17,21,22,34)(18,25,26,32)(19,28,29,45)(20,30,31,33)(23,36,37,43)(24,38,39,40)(27,41,42,44)(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,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,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, 107298);