#139052

order	512
nilpotency class	4
isoclinism	isoclinic class of #139013
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	#33752	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

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

normal subgroups only

This group has 412 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) = 64

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=gh,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=eghi,f^c=fh,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, 46248);