#126529

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₄²
commutator	C₄	C₄²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1726
prime radical	C₂³	#405
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

332 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 250 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2,3,6,4,5,7,8)(9,10,13,20,15,23,38,53,17,22,39,52,16,24,14,21)(11,18,26,41,28,44,61,71,30,43,62,70,29,45,27,42)(12,31,32,48,34,49,33,51,36,50,37,63,35,19,46,47)(25,54,55,66,57,67,56,69,59,68,60,72,58,40,64,65);(9,11,14,27,16,29,39,62,17,30,38,61,15,28,13,26)(10,18,21,42,24,45,52,70,22,43,53,71,23,44,20,41)(12,25,33,56,35,58,32,55,36,59,46,64,34,57,37,60)(19,40,48,66,50,68,47,65,49,67,63,72,31,54,51,69);(2,5)(6,8)(9,12)(10,19,22,49)(11,25)(13,37)(14,33)(15,34)(16,35)(17,36)(18,40,43,67)(20,51,52,47)(21,48,53,63)(23,31,24,50)(26,60)(27,56)(28,57)(29,58)(30,59)(32,39)(38,46)(41,69,70,65)(42,66,71,72)(44,54,45,68)(55,62)(61,64)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,g^2=i,h^2=i,c^a=cg,f^a=fi,d^c=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33725);