#198483

order	512
nilpotency class	3
isoclinism	isoclinic class of #198171
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

1096 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:

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

Polycyclic presentation:

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

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, 105679);