#199003

order	512
nilpotency class	3
isoclinism	isoclinic class of #198825
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	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

990 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dhi,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, 106199);