#199588

order	512
nilpotency class	3
isoclinism	isoclinic class of #199528
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1061 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)(3,11,13,7,14,15,12,16)(17,18)(19,28,32,39)(20,33)(21,26)(22,36)(23,27)(24,31,40,30)(25,34)(29,44,47,38)(35,41)(37,42)(43,46,48,45);(1,3)(2,7,10,16)(4,12)(5,13)(6,14)(8,15,9,11)(17,19,21,30,23,32,22,31)(18,24,26,39,27,40,36,28)(20,29,34,45,35,47,37,46)(25,38,41,48,42,44,33,43);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,20)(18,25)(19,29)(21,34)(22,37)(23,35)(24,38)(26,41)(27,42)(28,43)(30,45)(31,46)(32,47)(33,36)(39,48)(40,44);(1,4,6,5)(2,9,10,8)(3,12,14,13)(7,11,16,15)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,34,35,37)(24,39,40,28)(25,41,42,33)(29,45,47,46)(38,48,44,43)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106784);