#201906

order	512
nilpotency class	3
isoclinism	isoclinic class of #201717
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,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, 109102);