#305286

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

638 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 158 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)(9,10,13,15)(11,18,20,24)(12,14,21,17)(16,22,23,19)(27,34)(31,39)(35,40)(36,38);(10,14)(12,21)(15,17)(16,18)(19,22)(23,24)(25,26)(27,35)(28,32)(29,33)(30,37)(31,36)(34,40)(38,39);(1,3,4,8)(2,5,6,7)(10,15)(14,17)(16,23)(18,24)(25,27,29,34)(26,31,33,39)(28,36,30,38)(32,40,37,35);(9,11,13,20)(10,16,15,23)(12,22,21,19)(14,18,17,24)(25,28,29,30)(26,32,33,37)(27,36,34,38)(31,40,39,35)

Polycyclic presentation:

2^9|a^2=fi,c^2=fh,d^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gi,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212482);