#475496

order	512
nilpotency class	3
isoclinism	isoclinic class of #469927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2605
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17054
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

533 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 156 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)(3,15)(4,21)(5,11,29,42)(6,13)(7,12)(8,14)(9,18)(10,25)(16,31)(17,43,49,60)(19,34)(20,33)(22,35)(23,51,54,39)(24,38)(26,37)(27,55,53,36)(28,41,30,40)(32,50,61,48)(44,56,63,62)(45,57)(46,58)(47,64,52,59);(1,3,7,19,8,20,6,18)(2,9,13,33,14,34,12,15)(4,22,24,16,26,45,25,46)(5,17,30,50,29,49,28,48)(10,35,37,58,38,57,21,31)(11,32,41,60,42,61,40,43)(23,52,27,44,54,47,53,63)(36,59,39,62,55,64,51,56);(1,4,7,25,8,26,6,24)(2,10,13,21,14,38,12,37)(3,16,19,22,20,46,18,45)(5,27,28,54,29,53,30,23)(9,31,33,57,34,58,15,35)(11,39,40,55,42,51,41,36)(17,47,48,63,49,52,50,44)(32,59,43,62,61,64,60,56);(1,5,6,28,8,29,7,30)(2,11,13,41,14,42,12,40)(3,17,18,48,20,49,19,50)(4,23,25,53,26,54,24,27)(9,32,33,60,34,61,15,43)(10,36,37,39,38,55,21,51)(16,44,22,52,46,63,45,47)(31,56,35,59,58,62,57,64)

Polycyclic presentation:

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

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, 382692);