#219524

order	512
nilpotency class	3
isoclinism	isoclinic class of #219518
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

619 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 619 conjugacy classes of subgroups — too many to draw. Only its 178 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=efh,d^2=ehi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cei,f^b=fh,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126720);