#200832

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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

951 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)(4,17,20,25)(5,10)(6,11)(7,12)(8,15)(9,19,26,18)(14,28,30,22)(16,23)(21,24)(27,29,32,31)(33,34,38,43,39,42,37,50)(35,40,46,56,45,54,52,55)(36,47,49,41,48,58,51,57)(44,59,62,63,60,64,61,53);(1,3,5,15,7,16,6,21)(2,8,11,13,12,24,10,23)(4,14,19,31,20,30,18,29)(9,22,25,27,26,28,17,32)(33,35,38,52,39,45,37,46)(34,40,50,56,42,54,43,55)(36,44,49,61,48,60,51,62)(41,53,47,59,57,63,58,64);(1,4)(2,9)(3,14)(5,19)(6,18)(7,20)(8,22)(10,17)(11,25)(12,26)(13,27)(15,31)(16,30)(21,29)(23,32)(24,28)(33,36)(34,41)(35,44)(37,51)(38,49)(39,48)(40,53)(42,57)(43,58)(45,60)(46,62)(47,50)(52,61)(54,63)(55,64)(56,59);(1,5,7,6)(2,10,12,11)(3,15,16,21)(4,18,20,19)(8,23,24,13)(9,25,26,17)(14,29,30,31)(22,27,28,32)(34,42)(36,48)(40,54)(43,50)(44,60)(49,51)(55,56)(61,62)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,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, 108028);