#123223

order	512
nilpotency class	4
isoclinism	isoclinic class of #123211
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	#3186	C₂²
LCS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

296 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30419);