#138470

order	512
nilpotency class	4
isoclinism	isoclinic class of #138412
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	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

339 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45666);