#200180

order	512
nilpotency class	3
isoclinism	isoclinic class of #200122
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)(3,12,16,23,14,27,15,8)(4,17,19,25,20,9,18,24)(13,26,30,22,28,32,29,31)(33,34)(35,44,48,55)(36,49)(37,42)(38,52)(39,43)(40,47,56,46)(41,50)(45,60,63,54)(51,57)(53,58)(59,62,64,61);(1,3,7,14)(2,8)(4,13,20,28)(5,15,6,16)(9,22)(10,12)(11,23)(17,31)(18,29,19,30)(21,27)(24,32)(25,26)(33,35,37,46,39,48,38,47)(34,40,42,55,43,56,52,44)(36,45,50,61,51,63,53,62)(41,54,57,64,58,60,49,59);(1,4,7,20)(2,9,11,17)(3,13,14,28)(5,19,6,18)(8,22,23,31)(10,25,21,24)(12,26,27,32)(15,30,16,29)(33,36)(34,41)(35,45)(37,50)(38,53)(39,51)(40,54)(42,57)(43,58)(44,59)(46,61)(47,62)(48,63)(49,52)(55,64)(56,60);(3,14)(8,23)(12,27)(13,28)(15,16)(22,31)(26,32)(29,30)(33,37,39,38)(34,42,43,52)(35,46,48,47)(36,50,51,53)(40,55,56,44)(41,57,58,49)(45,61,63,62)(54,64,60,59)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=i,e^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gh

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, 107376);