#202041

order	512
nilpotency class	3
isoclinism	isoclinic class of #201958
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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1003 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=h,d^2=hi,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,f^b=fi,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, 109237);