#199543

order	512
nilpotency class	3
isoclinism	isoclinic class of #199540
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)(4,16,19,26)(5,20)(6,10)(7,11)(8,14)(9,18,25,17)(13,28,30,22)(15,23)(21,24)(27,29,32,31)(33,34)(35,39,45,53)(36,46,48,40)(37,49)(38,42)(41,50)(43,51)(44,57,58,52)(47,54,61,60)(55,63)(56,59)(62,64);(1,3,5,14,7,15,6,21)(2,8,10,12,11,24,20,23)(4,13,18,31,19,30,17,29)(9,22,26,27,25,28,16,32)(33,35)(34,39,42,53)(36,43)(37,44)(38,45)(40,51,46,59)(41,52,49,57)(47,55)(48,56)(50,58)(54,62,60,63)(61,64);(1,4)(2,9)(3,13)(5,18)(6,17)(7,19)(8,22)(10,26)(11,25)(12,27)(14,31)(15,30)(16,20)(21,29)(23,32)(24,28)(33,36)(34,40)(35,43)(37,47)(38,48)(39,51)(41,54)(42,46)(44,55)(45,56)(49,60)(50,61)(52,62)(53,59)(57,63)(58,64);(33,37,38,50)(34,41,42,49)(35,44,45,58)(36,47,48,61)(39,52,53,57)(40,54,46,60)(43,55,56,64)(51,62,59,63)

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,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, 106739);