#200903

order	512
nilpotency class	3
isoclinism	isoclinic class of #200746
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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

992 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=hi,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fh,g^c=ghi

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, 108099);