#120666

order	512
nilpotency class	3
isoclinism	isoclinic class of #10524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

446 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi

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, 27862);