#200154

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	#60953	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

981 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,f^2=i,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, 107350);