#199634

order	512
nilpotency class	3
isoclinism	isoclinic class of #199558
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	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106830);