#416414

order	512
nilpotency class	3
isoclinism	isoclinic class of #398772
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2439
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15223
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

887 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efh,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei

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, 323610);