#507969

order	512
nilpotency class	4
isoclinism	isoclinic class of #501330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13331	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

535 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 535 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,d^2=g,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415165);