#138533

order	512
nilpotency class	4
isoclinism	isoclinic class of #138482
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

395 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,6,10,14)(4,11,12,7)(5,8)(9,13)(15,16)(17,18,21,26,23,27,38,44,22,25,20,33,35,42,36,48)(19,28,31,41,32,43,37,40,30,46,29,45,39,47,34,24);(1,3,5,10)(2,6)(4,9,12,15)(7,13)(8,14)(11,16)(17,19,22,30)(18,24)(20,34,21,37)(23,39,35,32)(25,40)(26,43)(27,45)(28,44)(29,38,31,36)(33,47)(41,42)(46,48);(1,4,5,12)(2,7,8,11)(3,9,10,15)(6,13,14,16)(18,25)(20,21)(24,40)(27,42)(29,31)(34,37)(36,38)(41,45)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=gi,d^2=ghi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fh,h^b=hi,d^c=dg,f^c=fg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45729);