#199386

order	512
nilpotency class	3
isoclinism	isoclinic class of #199221
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	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1079 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,22,27,23,26,21,35)(19,24,30,41,29,39,37,40)(20,31,33,25,34,44,36,43)(28,45,47,42,32,48,46,38);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(17,19,22,37,23,29,21,30)(18,24,35,41,26,39,27,40)(20,32,33,47,34,28,36,46)(25,42,31,48,43,38,44,45);(3,11)(7,15)(10,16)(12,13)(17,20)(18,25)(19,28)(21,36)(22,33)(23,34)(24,38)(26,43)(27,44)(29,32)(30,47)(31,35)(37,46)(39,42)(40,48)(41,45);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(18,26)(24,39)(25,43)(27,35)(31,44)(38,42)(40,41)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,e^b=ei,g^b=gi,f^c=fi

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, 106582);