#503524

order	512
nilpotency class	4
isoclinism	isoclinic class of #501058
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×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 826 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)(12,27)(13,18)(14,23)(15,34)(16,21)(17,22)(19,31)(20,26)(25,37)(28,36)(29,38)(30,40)(32,39)(33,35);(1,3,5,4)(2,6,8,7)(9,11,14,13,17,26,16,15)(10,18,21,24,22,34,23,20)(12,28,30,31,32,25,33,29)(19,35,36,39,38,40,37,27);(1,3,5,4)(2,7,8,6)(9,12,11,25,16,33,15,31,17,32,26,28,14,30,13,29)(10,19,18,35,23,37,20,27,22,38,34,40,21,36,24,39);(1,4,5,3)(2,7,8,6)(9,13,14,26,17,15,16,11)(10,20,21,18,22,24,23,34)(12,29,30,28,32,31,33,25)(19,27,36,35,38,39,37,40)

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,d^b=di,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, 410720);