#237926

order	512
nilpotency class	3
isoclinism	isoclinic class of #233668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

820 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=i,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=ghi,f^c=fh,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, 145122);