#142712

order	512
nilpotency class	4
isoclinism	isoclinic class of #142632
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	#33758	C₂
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

432 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 70 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)(19,31)(20,37)(21,26)(22,29)(23,42)(24,30)(25,33)(27,39)(28,40)(32,47)(34,46)(35,43)(36,44)(38,48)(41,45);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(17,19,24,36)(18,25,30,46)(20,38,21,41)(22,32,23,35)(26,44,37,31)(27,43,28,47)(29,48,42,45)(33,40,34,39);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(18,26)(19,32)(20,39)(21,40)(22,23)(25,43)(27,42)(28,29)(30,37)(31,45)(33,34)(35,36)(44,48)(46,47)

Polycyclic presentation:

2^9|b^2=gh,d^2=gi,e^2=gi,f^2=i,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,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=eg,f^c=fi,g^c=gi,h^c=hi

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, 49908);