#198788

order	512
nilpotency class	3
isoclinism	isoclinic class of #9279
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1006 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105984);