#200366

order	512
nilpotency class	3
isoclinism	isoclinic class of #200146
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

1018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,14,19,40,4,24)(3,17,22,41,49,43,20,33)(5,28,31,45,50,13,27,37)(7,15,25,9,8,16,26,12)(10,35,32,30,44,56,34,29)(11,38,39,18,46,48,55,23)(21,52,54,59,64,36,51,58)(42,60,62,53,57,63,61,47);(2,9,15,14)(3,18)(4,25)(5,29)(6,26)(10,32,28,45)(11,39,43,33)(12,16,24,40)(13,37,44,34)(17,41,46,55)(20,22)(23,49)(27,31)(30,50)(35,56)(36,61,57,58)(38,48)(42,59,52,62)(51,60)(54,63);(2,10)(4,26)(5,30)(6,25)(9,32)(11,36)(12,34)(13,16)(14,45)(15,28)(17,42)(20,38)(21,53)(22,48)(24,37)(27,31)(29,50)(33,58)(35,56)(39,61)(40,44)(41,59)(43,57)(46,52)(47,64)(51,54)(55,62)(60,63);(1,3,19,49)(2,11,40,46)(4,20,6,22)(5,21,50,64)(7,18,8,23)(9,33,12,41)(10,36,44,52)(13,42,28,57)(14,39,24,55)(15,43,16,17)(25,48,26,38)(27,51,31,54)(29,47,30,53)(32,58,34,59)(35,60,56,63)(37,62,45,61)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=i,g^2=hi,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,e^c=eh,f^c=fi,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, 107562);