#343695

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
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

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

normal subgroups only

This group has 498 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)(9,10,13,17,14,18,11,15)(12,20,19,16,21,23,22,24)(27,34)(31,39)(35,40)(36,38);(1,3)(2,6)(4,7)(5,8)(9,11,14,13)(10,15,18,17)(12,19,21,22)(16,23,24,20)(25,26,29,33)(27,35,34,40)(28,32,30,37)(31,38,39,36);(1,4,5,6)(2,3,7,8)(10,15)(11,13)(16,23)(17,18)(19,22)(20,24)(25,27,29,34)(26,31,33,39)(28,36,30,38)(32,40,37,35);(2,7)(3,8)(9,12,14,21)(10,16,18,24)(11,19,13,22)(15,23,17,20)(25,28,29,30)(26,32,33,37)(27,36,34,38)(31,40,39,35)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=ei,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, 250891);