#201066

order	512
nilpotency class	3
isoclinism	isoclinic class of #200938
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

1008 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=h,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=fi,g^b=gi,e^c=ei,f^c=fh,g^c=gi

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, 108262);