#345337

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
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

535 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 535 conjugacy classes of subgroups — too many to draw. Only its 144 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=eh,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,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, 252533);