#503564

order	512
nilpotency class	4
isoclinism	isoclinic class of #501248
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	#33582	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 750 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)(9,10,13,19)(11,22,23,17)(12,15,28,32)(14,30,31,20)(16,29,18,21)(24,27,40,36)(25,39,33,35)(26,37,34,38);(1,3,5,4)(2,6,8,7)(9,11,13,23)(10,17,19,22)(12,27,28,36)(14,33,31,25)(15,24,32,40)(16,34,18,26)(20,35,30,39)(21,38,29,37);(1,3,5,4)(2,7,8,6)(9,12,15,19,16,30,14,29,13,28,32,10,18,20,31,21)(11,22,33,36,26,38,24,39,23,17,25,27,34,37,40,35);(1,4,5,3)(2,7,8,6)(10,19)(12,28)(17,22)(20,30)(21,29)(27,36)(35,39)(37,38)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=efh,d^2=ei,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi,d^c=di

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, 410760);