#200313

order	512
nilpotency class	3
isoclinism	isoclinic class of #200158
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	#60949	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1071 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:

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

Polycyclic presentation:

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

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, 107509);