#343559

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 496 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) = 512

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

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

Polycyclic presentation:

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

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, 250755);