#201267

order	512
nilpotency class	3
isoclinism	isoclinic class of #201121
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	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

973 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)(3,12)(4,17,20,9)(5,10)(6,21)(7,11)(8,15)(13,26,30,32)(14,23)(16,27)(18,25,19,24)(22,29,31,28)(33,34,38,44,39,42,37,43)(35,45,49,40,48,56,47,55)(36,50,52,57,53,41,51,58)(46,59,62,63,61,64,60,54);(1,3,5,14,7,16,6,15)(2,8,21,27,11,23,10,12)(4,13,19,29,20,30,18,28)(9,22,24,32,17,31,25,26)(33,35)(34,40)(36,46)(37,49)(38,47)(39,48)(41,54)(42,55)(43,56)(44,45)(50,63)(51,62)(52,60)(53,61)(57,59)(58,64);(1,4)(2,9)(3,13)(5,19)(6,18)(7,20)(8,22)(10,25)(11,17)(12,26)(14,29)(15,28)(16,30)(21,24)(23,31)(27,32)(33,36,39,53)(34,41,42,50)(35,46,48,61)(37,52,38,51)(40,54,55,63)(43,58,44,57)(45,59,56,64)(47,62,49,60);(1,5,7,6)(2,10,11,21)(3,14,16,15)(4,18,20,19)(8,12,23,27)(9,24,17,25)(13,28,30,29)(22,32,31,26)(34,42)(40,55)(41,50)(43,44)(45,56)(54,63)(57,58)(59,64)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,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, 108463);