#507600

order	512
nilpotency class	4
isoclinism	isoclinic class of #500891
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	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

414 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 144 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)(3,7,8,6)(9,10)(11,16)(12,17)(13,19)(14,28)(15,20)(18,26)(21,35)(22,30)(23,32)(24,31)(25,33)(27,34)(29,36)(37,40)(38,39);(2,5)(6,7)(9,11,13,24,15,22,14,21)(10,16,19,31,20,30,28,35)(12,23,25,38,27,37,26,36)(17,32,33,39,34,40,18,29);(10,17,20,34)(11,21)(12,25)(13,14)(16,29,30,39)(18,28,33,19)(22,24)(26,27)(31,40,35,32)(36,38);(1,3,4,8)(2,6,5,7)(11,22)(16,30)(21,24)(23,37)(29,39)(31,35)(32,40)(36,38)

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=gh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fhi,g^c=gi,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, 414796);