#266713

order	512
nilpotency class	3
isoclinism	isoclinic class of #246799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

644 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 148 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=hi,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,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, 173909);