#345661

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

512 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 146 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)(9,10,15,23,14,19,13,18)(11,16,22,24,12,21,20,17)(25,26,29,31)(27,34,36,32)(28,37,30,33)(35,39,38,40);(1,3,5,8)(2,6,7,4)(9,11,14,12)(10,16,19,21)(13,20,15,22)(17,23,24,18)(26,31)(27,35)(28,30)(32,39)(34,40)(36,38);(9,12,14,11)(10,17,19,24)(13,20,15,22)(16,23,21,18)(25,27,29,36)(26,32,31,34)(28,35,30,38)(33,39,37,40);(1,4,5,6)(2,3,7,8)(10,18)(13,15)(16,17)(19,23)(20,22)(21,24)(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=hi,d^2=e,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, 252857);