#345582

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

544 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,c^2=hi,d^2=eh,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, 252778);