#507581

order	512
nilpotency class	4
isoclinism	isoclinic class of #500599
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 511 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=ei,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fh,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, 414777);