#199559

order	512
nilpotency class	3
isoclinism	isoclinic class of #199558
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)(3,13)(4,18,21,25)(5,10)(6,11)(7,12)(8,16)(9,20,26,19)(14,28,31,22)(15,24)(17,23)(27,30,32,29)(33,34)(35,39,45,55)(36,46,48,40)(37,49)(38,42)(41,52)(43,53)(44,57,50,54)(47,56,51,58)(59,64)(60,62)(61,63);(1,3,5,16,7,17,6,15)(2,8,11,13,12,24,10,23)(4,14,20,29,21,31,19,30)(9,22,25,27,26,28,18,32)(33,35)(34,39,42,55)(36,43)(37,50)(38,45)(40,53,46,62)(41,57,49,54)(44,52)(47,61)(48,60)(51,59)(56,64,58,63);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,22)(10,18)(11,25)(12,26)(13,27)(15,30)(16,29)(17,31)(23,32)(24,28)(33,36)(34,40)(35,43)(37,51)(38,48)(39,53)(41,58)(42,46)(44,61)(45,60)(47,52)(49,56)(50,59)(54,64)(55,62)(57,63);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,24,23)(9,25,26,18)(14,29,31,30)(22,32,28,27)(33,37,38,52)(34,41,42,49)(35,44,45,50)(36,47,48,51)(39,54,55,57)(40,56,46,58)(43,59,60,61)(53,63,62,64)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,f^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,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, 106755);