#345519

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

622 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=ehi,c^2=ehi,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defhi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

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, 252715);