#139022

order	512
nilpotency class	4
isoclinism	isoclinic class of #139022
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	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 296 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) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=ghi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=egh,f^c=fhi,h^c=hi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46218);