#105031

order	512
nilpotency class	3
isoclinism	isoclinic class of #105009
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

392 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=d,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,c^b=cf,d^b=dg,e^b=ei,e^c=egh,f^c=fg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12227);