#129121

order	512
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#442
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8275
prime radical	C₂³	#430
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

237 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 140 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,d^2=gi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=cg,d^a=di,d^b=dg,e^b=egi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36317);