#143569

order	512
nilpotency class	4
isoclinism	isoclinic class of #143518
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

328 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50765);