#139040

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	#33779	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 331 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,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, 46236);