#200816

order	512
nilpotency class	3
isoclinism	isoclinic class of #200737
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)(4,16,19,26)(5,20)(6,10)(7,11)(8,15)(9,18,25,17)(13,28,29,22)(14,24)(21,23)(27,30,32,31)(33,34,38,43,39,42,37,50)(35,40,46,55,45,56,52,54)(36,47,48,41,49,58,51,57)(44,59,62,63,60,64,61,53);(1,3,6,21,7,14,5,15)(2,8,20,24,11,23,10,12)(4,13,17,30,19,29,18,31)(9,22,16,32,25,28,26,27)(33,35,37,46,39,45,38,52)(34,40,43,54,42,56,50,55)(36,44,51,62,49,60,48,61)(41,53,58,64,57,63,47,59);(1,4)(2,9)(3,13)(5,18)(6,17)(7,19)(8,22)(10,26)(11,25)(12,27)(14,29)(15,31)(16,20)(21,30)(23,28)(24,32)(33,36,39,49)(34,41,42,57)(35,44,45,60)(37,51,38,48)(40,53,56,63)(43,58,50,47)(46,62,52,61)(54,64,55,59);(3,14)(8,23)(12,24)(13,29)(15,21)(22,28)(27,32)(30,31)(34,42)(35,45)(41,57)(43,50)(44,60)(46,52)(47,58)(61,62)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,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,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,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, 108012);