#197088

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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1007 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,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, 104284);