#198812

order	512
nilpotency class	3
isoclinism	isoclinic class of #198807
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1080 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)(3,6,10,15)(4,7)(5,8)(9,12)(11,13,16,14)(17,18)(19,28)(20,32,34,25)(21,26)(22,36)(23,27)(24,30)(29,43,46,48)(31,39)(33,42,37,41)(35,40)(38,45,47,44);(1,3)(2,6,8,15)(4,11)(5,10)(7,13,9,14)(12,16)(17,19,22,30,23,31,21,35)(18,24,26,28,27,40,36,39)(20,29,37,44,34,46,33,45)(25,38,41,43,32,47,42,48);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,25)(19,29)(21,33)(22,37)(23,34)(24,38)(26,41)(27,32)(28,43)(30,44)(31,46)(35,45)(36,42)(39,48)(40,47);(2,8)(3,10)(7,9)(11,16)(17,21,23,22)(18,26,27,36)(19,30,31,35)(20,33,34,37)(24,39,40,28)(25,41,32,42)(29,44,46,45)(38,48,47,43)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,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, 106008);