#200942

order	512
nilpotency class	3
isoclinism	isoclinic class of #200938
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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

970 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)(3,13)(4,18)(5,10)(6,11)(7,12)(8,16)(9,19)(14,28)(15,24)(17,23)(20,25)(21,26)(22,31)(27,30)(29,32)(33,34)(35,44,47,55)(36,48,50,41)(37,51)(38,42)(39,43)(40,46,56,52)(45,59)(49,57,53,58)(54,61)(60,63)(62,64);(1,3,5,16,7,17,6,15)(2,8,11,13,12,24,10,23)(4,14,19,30,20,31,21,29)(9,22,18,27,26,28,25,32)(33,35,38,52,39,47,37,46)(34,40,42,55,43,56,51,44)(36,45,53,61,50,62,49,60)(41,54,57,64,48,63,58,59);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,25)(11,18)(12,26)(13,27)(15,29)(16,30)(17,31)(23,32)(24,28)(33,36)(34,41)(35,45)(37,49)(38,53)(39,50)(40,54)(42,57)(43,48)(44,59)(46,60)(47,62)(51,58)(52,61)(55,64)(56,63);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,20,21)(8,13,24,23)(9,25,26,18)(14,29,31,30)(22,27,28,32)(33,37,39,38)(34,42,43,51)(35,46,47,52)(36,49,50,53)(40,55,56,44)(41,57,48,58)(45,60,62,61)(54,64,63,59)

Polycyclic presentation:

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

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, 108138);