#142708

order	512
nilpotency class	4
isoclinism	isoclinic class of #142624
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

428 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 76 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)(9,12)(10,13)(11,15)(14,16)(17,18,22,27)(19,28,31,46)(20,24,35,42)(21,36,34,26)(23,38,40,29)(25,43,41,33)(30,47,37,44)(32,45,48,39);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(17,19,22,31)(18,23,27,40)(20,32,35,48)(21,37,34,30)(24,28,42,46)(25,44,41,47)(26,45,36,39)(29,43,38,33);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(18,24)(20,33)(21,34)(23,28)(25,26)(27,42)(29,32)(30,37)(35,43)(36,41)(38,48)(39,47)(40,46)(44,45)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gi,e^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gi,h^b=hi,d^c=dh,e^c=eg,g^c=gi,h^c=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49904);