#199376

order	512
nilpotency class	3
isoclinism	isoclinic class of #199212
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

1145 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)(7,12)(11,15)(13,16)(17,18,22,27,23,26,21,34)(19,24,30,40,29,38,36,39)(20,31,32,25,33,42,35,41)(28,43,46,47,44,48,45,37);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(17,19,22,36,23,29,21,30)(18,24,34,40,26,38,27,39)(20,28,32,45,33,44,35,46)(25,37,31,43,41,47,42,48);(17,20)(18,25)(19,28)(21,35)(22,32)(23,33)(24,37)(26,41)(27,42)(29,44)(30,46)(31,34)(36,45)(38,47)(39,48)(40,43);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(18,26)(24,38)(25,41)(27,34)(31,42)(37,47)(39,40)(43,48)

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,e^b=ei,g^b=gi,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106572);