#123126

order	512
nilpotency class	4
isoclinism	isoclinic class of #123111
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

304 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 42 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)(3,9)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(17,18,23,27)(19,28,32,46)(20,25,22,38)(21,34,37,45)(24,39,41,30)(26,33,44,36)(29,43,31,48)(35,42,47,40);(3,10)(6,15)(7,12)(11,13)(17,19,23,32)(18,24,27,41)(20,31,22,29)(21,35,37,47)(25,42,38,40)(26,43,44,48)(28,45,46,34)(30,33,39,36);(1,3,5,10)(2,6,8,15)(4,13,14,11)(7,16,12,9)(17,20,23,22)(18,25,27,38)(19,29,32,31)(21,36,37,33)(24,40,41,42)(26,34,44,45)(28,43,46,48)(30,35,39,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=fh,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=ehi,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=ehi,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30322);