#200817

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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

951 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)(33,34,38,42,39,43,37,52)(35,40,47,54,46,56,45,55)(36,48,50,41,51,57,49,58)(44,59,62,63,61,64,60,53);(1,3,5,15,7,14,6,21)(2,8,10,12,11,23,20,24)(4,13,18,31,19,29,17,30)(9,22,26,27,25,28,16,32)(33,35,38,45,39,46,37,47)(34,40,52,54,43,56,42,55)(36,44,50,60,51,61,49,62)(41,53,48,59,58,63,57,64);(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)(34,41)(35,44)(37,49)(38,50)(39,51)(40,53)(42,57)(43,58)(45,60)(46,61)(47,62)(48,52)(54,59)(55,64)(56,63);(3,14)(8,23)(12,24)(13,29)(15,21)(22,28)(27,32)(30,31)(33,37,39,38)(34,42,43,52)(35,45,46,47)(36,49,51,50)(40,54,56,55)(41,57,58,48)(44,60,61,62)(53,59,63,64)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^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, 108013);