#200373

order	512
nilpotency class	3
isoclinism	isoclinic class of #200152
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107569);