#122044

order	512
nilpotency class	3
isoclinism	isoclinic class of #120247
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#391
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1446
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

658 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=hi,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=dh,g^b=gi,d^c=di

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, 29240);