#200222

order	512
nilpotency class	3
isoclinism	isoclinic class of #200152
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

939 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^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=fhi,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fi,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, 107418);