#470885

order	512
nilpotency class	3
isoclinism	isoclinic class of #462392
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2517
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16707
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

702 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378081);