#200826

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	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

936 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)(4,16,17,26)(5,20)(6,10)(7,11)(8,14)(9,19,25,18)(13,28,29,22)(15,23)(21,24)(27,30,32,31)(33,34,38,43,39,42,37,50)(35,40,46,56,45,54,52,55)(36,47,49,41,48,58,51,57)(44,59,62,63,60,64,61,53);(1,3,5,14,7,15,6,21)(2,8,10,12,11,24,20,23)(4,13,19,31,17,29,18,30)(9,22,26,27,25,28,16,32)(33,35,38,52,39,45,37,46)(34,40,50,56,42,54,43,55)(36,44,49,61,48,60,51,62)(41,53,47,59,57,63,58,64);(1,4)(2,9)(3,13)(5,19)(6,18)(7,17)(8,22)(10,26)(11,25)(12,27)(14,31)(15,29)(16,20)(21,30)(23,32)(24,28)(33,36,39,48)(34,41,42,57)(35,44,45,60)(37,51,38,49)(40,53,54,63)(43,58,50,47)(46,62,52,61)(55,64,56,59);(4,17)(9,25)(13,29)(16,26)(18,19)(22,28)(27,32)(30,31)(34,42)(36,48)(40,54)(43,50)(44,60)(49,51)(55,56)(61,62)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,e^2=hi,f^2=h,g^2=i,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, 108022);