#199034

order	512
nilpotency class	3
isoclinism	isoclinic class of #198843
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	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

964 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)(4,15,17,8)(5,18)(6,10)(7,14)(9,20)(12,25,27,32)(13,26)(16,23,19,24)(21,28,31,30)(22,29)(33,34,38,43,39,42,37,50)(35,40,46,56,45,54,52,55)(36,47,48,41,49,58,51,57)(44,59,62,63,60,64,61,53);(1,3,5,14,6,13,20,29)(2,7,9,11,10,22,18,26)(4,12,19,30,17,27,16,28)(8,21,24,25,15,31,23,32)(33,35,38,52,39,45,37,46)(34,40,50,56,42,54,43,55)(36,44,48,61,49,60,51,62)(41,53,47,59,57,63,58,64);(1,4)(2,8)(3,12)(5,19)(6,17)(7,21)(9,24)(10,15)(11,25)(13,27)(14,30)(16,20)(18,23)(22,31)(26,32)(28,29)(33,36,39,49)(34,41,42,57)(35,44,45,60)(37,51,38,48)(40,53,54,63)(43,58,50,47)(46,62,52,61)(55,64,56,59);(3,13)(7,22)(11,26)(12,27)(14,29)(21,31)(25,32)(28,30)(34,42)(40,54)(41,57)(43,50)(47,58)(53,63)(55,56)(59,64)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^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=fh,d^b=dhi,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi

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, 106230);