#503521

order	512
nilpotency class	4
isoclinism	isoclinic class of #501060
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

796 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410717);