#139054

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	C₂×#466	C₂²
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

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

normal subgroups only

This group has 399 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)(17,18,22,28,38,47,37,27,24,30,20,26,23,29,21,39)(19,31,34,45,41,43,33,25,36,44,32,48,35,42,40,46);(2,7,10,9)(8,11,16,15)(17,19)(18,25)(20,33)(21,34)(22,40)(23,41)(24,36)(26,42)(27,31)(28,43)(29,48)(30,46)(32,37)(35,38)(39,44)(45,47);(1,3)(2,8)(4,12)(5,13)(6,14)(7,11)(9,15)(10,16)(18,26)(20,37)(21,22)(23,38)(25,42)(27,47)(28,30)(29,39)(31,45)(32,33)(34,40)(35,41)(43,46)(44,48)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,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, 46250);