#312241

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
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

546 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=hi,c^2=h,d^2=efi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi,e^d=eh,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 219437);