#233333

order	512
nilpotency class	3
isoclinism	isoclinic class of #233331
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1278 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1278 conjugacy classes of subgroups — too many to draw. Only its 198 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,e^b=eh,g^b=ghi,d^c=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140529);