#198909

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	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

946 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,e^2=hi,f^2=i,g^2=h,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, 106105);