#144258

order	512
nilpotency class	4
isoclinism	isoclinic class of #144232
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	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 423 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51454);