#200755

order	512
nilpotency class	3
isoclinism	isoclinic class of #200746
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	#60972	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

973 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)(4,16,17,26)(5,20)(6,10)(7,11)(8,15)(9,19,25,18)(13,28,30,22)(14,24)(21,23)(27,29,32,31)(33,34)(35,43,46,57)(36,47,49,40)(37,42)(38,41)(39,45,54,50)(44,59)(48,58,51,55)(52,56)(53,60)(61,64)(62,63);(1,3,5,15,7,14,6,21)(2,8,10,12,11,23,20,24)(4,13,19,31,17,30,18,29)(9,22,26,27,25,28,16,32)(33,35,37,50,38,46,52,45)(34,39,42,57,41,54,56,43)(36,44,51,60,49,61,48,62)(40,53,58,64,47,63,55,59);(1,4)(2,9)(3,13)(5,19)(6,18)(7,17)(8,22)(10,26)(11,25)(12,27)(14,30)(15,31)(16,20)(21,29)(23,28)(24,32)(33,36)(34,40)(35,44)(37,51)(38,49)(39,53)(41,47)(42,58)(43,59)(45,62)(46,61)(48,52)(50,60)(54,63)(55,56)(57,64);(3,14)(4,17)(8,23)(9,25)(12,24)(15,21)(16,26)(18,19)(34,41)(39,54)(40,47)(42,56)(43,57)(53,63)(55,58)(59,64)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=hi,f^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fh,g^c=ghi

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, 107951);