#139045

order	512
nilpotency class	4
isoclinism	isoclinic class of #139008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#33754	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

375 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,d^2=ghi,e^2=ghi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=egh,f^c=fh,h^c=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46241);