#121668

order	512
nilpotency class	3
isoclinism	isoclinic class of #111436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

460 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,6,4,5,7,8)(9,10)(11,20,24,36)(12,25)(13,18)(14,19)(15,29)(16,22,32,23)(17,28)(21,37,38,33)(26,40,39,31)(27,35)(30,34);(2,5)(3,7)(9,11,15,22,14,24,13,23)(10,16,29,20,19,32,18,36)(12,26,30,38,28,39,27,21)(17,33,35,31,25,37,34,40);(2,5)(3,7)(9,12,15,30,14,28,13,27)(10,17,29,35,19,25,18,34)(11,21,22,26,24,38,23,39)(16,31,20,37,32,40,36,33)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=ghi,f^c=fi,g^c=gh

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, 28864);