#317709

order	512
nilpotency class	3
isoclinism	isoclinic class of #262883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

508 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,d^b=dh,f^c=fh,g^c=gi

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, 224905);