#200310

order	512
nilpotency class	3
isoclinism	isoclinic class of #200158
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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

985 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)(3,11)(4,15,16,8)(5,18)(6,10)(7,14)(9,20)(12,25,28,32)(13,26)(17,23,19,24)(21,27,31,30)(22,29)(33,34,38,42,39,43,37,44)(35,40,48,54,47,56,46,55)(36,49,52,41,51,57,50,58)(45,59,61,53,62,63,60,64);(1,3,5,14,6,13,20,29)(2,7,9,11,10,22,18,26)(4,12,19,30,16,28,17,27)(8,21,24,25,15,31,23,32)(33,35,38,46,39,47,37,48)(34,40,44,54,43,56,42,55)(36,45,50,61,51,62,52,60)(41,53,57,59,58,64,49,63);(1,4)(2,8)(3,12)(5,19)(6,16)(7,21)(9,24)(10,15)(11,25)(13,28)(14,30)(17,20)(18,23)(22,31)(26,32)(27,29)(33,36)(34,41)(35,45)(37,52)(38,50)(39,51)(40,53)(42,49)(43,58)(44,57)(46,61)(47,62)(48,60)(54,59)(55,63)(56,64);(3,13)(4,16)(7,22)(8,15)(11,26)(14,29)(17,19)(23,24)(33,37,39,38)(34,42,43,44)(35,46,47,48)(36,50,51,52)(40,54,56,55)(41,57,58,49)(45,60,62,61)(53,63,64,59)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fi,g^c=gh

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, 107506);