#505727

order	512
nilpotency class	4
isoclinism	isoclinic class of #500112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

524 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 128

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

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

Polycyclic presentation:

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

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, 412923);