#199785

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	#60949	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1114 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=i,e^2=hi,f^2=hi,g^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, 106981);