#197083

order	512
nilpotency class	3
isoclinism	isoclinic class of #197020
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

993 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,e^2=hi,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=ei,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh,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, 104279);