#197391

order	512
nilpotency class	3
isoclinism	isoclinic class of #197339
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	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1090 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,22,27,23,28,21,26)(19,29,33,40,32,39,31,24)(20,34,36,41,37,25,35,42)(30,43,46,38,45,48,44,47);(1,3,4,12,5,11,13,10)(2,6,8,16,7,14,15,9)(17,19,23,32)(18,24)(20,30,37,45)(21,31,22,33)(25,38)(26,39)(27,29)(28,40)(34,47)(35,44,36,46)(41,43)(42,48);(17,20,23,37)(18,25,28,34)(19,30,32,45)(21,36,22,35)(24,38,40,47)(26,42,27,41)(29,43,39,48)(31,46,33,44);(2,7)(6,14)(8,15)(9,16)(17,21,23,22)(18,26,28,27)(19,31,32,33)(20,35,37,36)(24,39,40,29)(25,41,34,42)(30,44,45,46)(38,43,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=h,e^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,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, 104587);