#199020

order	512
nilpotency class	3
isoclinism	isoclinic class of #198834
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)(3,12)(4,17,19,9)(5,10)(6,20)(7,11)(8,15)(13,26,29,32)(14,23)(16,27)(18,25,21,24)(22,28,31,30)(33,34,38,44,39,42,37,43)(35,45,49,40,47,55,48,56)(36,50,52,41,53,58,51,57)(46,54,62,59,60,64,61,63);(1,3,6,15,7,16,5,14)(2,8,10,12,11,23,20,27)(4,13,21,30,19,29,18,28)(9,22,24,26,17,31,25,32)(33,35)(34,40)(36,46)(37,49)(38,48)(39,47)(41,54)(42,56)(43,55)(44,45)(50,59)(51,62)(52,61)(53,60)(57,64)(58,63);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,24)(11,17)(12,26)(14,28)(15,30)(16,29)(20,25)(23,31)(27,32)(33,36,39,53)(34,41,42,57)(35,46,47,60)(37,52,38,51)(40,54,56,64)(43,58,44,50)(45,59,55,63)(48,62,49,61);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,18,19,21)(8,12,23,27)(9,24,17,25)(13,28,29,30)(22,26,31,32)(34,42)(35,47)(41,57)(43,44)(46,60)(48,49)(50,58)(61,62)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=h,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=di,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, 106216);