#345650

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

512 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

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

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, 252846);