#508803

order	512
nilpotency class	4
isoclinism	isoclinic class of #508669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13341	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

518 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Polycyclic presentation:

2^9|a^2=e,d^2=efh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,e^b=ei,g^b=gh,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gh,h^c=hi,g^d=gi

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, 415999);