#412543

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	#15131
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

943 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,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, 319739);