#199555

order	512
nilpotency class	3
isoclinism	isoclinic class of #199552
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

969 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)(35,39,45,54)(36,46,48,40)(37,49)(38,42)(41,51)(43,52)(44,59,61,53)(47,55,50,56)(57,64)(58,62)(60,63);(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)(34,39,42,54)(36,43)(37,44)(38,45)(40,52,46,62)(41,53,49,59)(47,57)(48,58)(50,60)(51,61)(55,63,56,64);(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)(34,40)(35,43)(37,50)(38,48)(39,52)(41,56)(42,46)(44,60)(45,58)(47,51)(49,55)(53,64)(54,62)(57,61)(59,63);(4,17)(9,25)(13,29)(16,26)(18,19)(22,28)(27,32)(30,31)(33,37,38,51)(34,41,42,49)(35,44,45,61)(36,47,48,50)(39,53,54,59)(40,55,46,56)(43,57,58,60)(52,63,62,64)

Polycyclic presentation:

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

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, 106751);