#198807

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	#60965	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1091 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)(19,27)(20,31,33,24)(21,34)(22,26)(23,30)(25,36)(28,41,43,48)(29,42)(32,39,35,40)(37,44,47,46)(38,45);(1,3)(2,6,8,15)(4,11)(5,10)(7,13,9,14)(12,16)(17,19,21,30,22,29,36,45)(18,23,25,27,26,38,34,42)(20,28,35,46,33,43,32,44)(24,37,40,41,31,47,39,48);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,24)(19,28)(21,35)(22,33)(23,37)(25,40)(26,31)(27,41)(29,43)(30,46)(32,36)(34,39)(38,47)(42,48)(44,45);(2,8)(3,10)(7,9)(11,16)(19,29)(23,38)(27,42)(28,43)(30,45)(37,47)(41,48)(44,46)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,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, 106003);