#508633

order	512
nilpotency class	4
isoclinism	isoclinic class of #508579
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

425 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 138 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)(9,10,16,20,13,19,14,15)(11,21,33,31,23,17,24,36)(12,25,30,18,27,32,28,29)(22,38,40,37,35,26,39,34);(2,5)(6,8)(9,11)(10,17)(12,26)(13,23)(14,24)(15,31)(16,33)(18,35)(19,21)(20,36)(22,29)(25,40)(27,38)(28,37)(30,34)(32,39);(3,7)(6,8)(9,12)(10,18)(11,22,23,35)(13,27)(14,30)(15,32)(16,28)(17,34,21,37)(19,29)(20,25)(24,40,33,39)(26,31,38,36);(1,3,4,7)(2,6,5,8)(12,27)(18,29)(22,35)(25,32)(26,38)(28,30)(34,37)(39,40)

Polycyclic presentation:

2^9|a^2=ef,c^2=fh,d^2=ei,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,d^c=de,f^c=fi,g^c=ghi,h^c=hi

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, 415829);