#201021

order	512
nilpotency class	3
isoclinism	isoclinic class of #200944
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	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

947 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)(3,13)(4,18)(5,10)(6,11)(7,12)(8,15)(9,20)(14,28)(16,24)(17,23)(19,25)(21,26)(22,31)(27,30)(29,32)(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);(1,3,6,16,7,17,5,15)(2,8,10,23,12,24,11,13)(4,14,19,29,21,31,20,30)(9,22,26,32,25,28,18,27)(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)(3,14)(5,20)(6,19)(7,21)(8,22)(10,26)(11,18)(12,25)(13,27)(15,30)(16,29)(17,31)(23,32)(24,28)(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)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,23,24,13)(9,18,25,26)(14,29,31,30)(22,27,28,32)(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,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,e^b=ei,f^b=fi,g^b=gi,d^c=dh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108217);