#198250

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	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

1050 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:

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

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,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, 105446);