#199560

order	512
nilpotency class	3
isoclinism	isoclinic class of #199558
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	C₂³×C₄	C₂⁴
prime radical	#60951	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1039 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,e^2=hi,f^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fhi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106756);