#123123

order	512
nilpotency class	4
isoclinism	isoclinic class of #123111
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	#31736	C₂
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

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

normal subgroups only

This group has 360 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,6)(4,7)(5,9)(8,13)(10,15)(11,16)(12,14)(17,18,22,26)(19,24,21,30)(20,23,29,32)(25,27,31,28);(1,3)(2,6)(4,12)(5,10)(7,16)(8,14)(9,15)(11,13)(18,23)(19,21)(20,28)(24,25)(26,32)(27,29)(30,31);(2,7)(3,10)(4,13)(6,14)(8,9)(15,16)(17,19,22,21)(18,24,26,30)(20,28,29,27)(23,31,32,25)

Polycyclic presentation:

2^9|a^2=h,c^2=h,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, 30319);