#309583

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
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

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

normal subgroups only

This group has 524 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)(3,6)(4,5)(7,8)(9,10,14,19)(11,16,12,21)(13,17,23,18)(15,22,24,20)(25,26,30,29)(27,31,34,37)(28,32,36,38)(33,39,40,35);(9,11,14,12)(10,15,19,24)(13,22,23,20)(16,18,21,17)(25,27,30,34)(26,31,29,37)(28,35,36,39)(32,33,38,40);(2,5)(6,8)(9,12,14,11)(10,16,19,21)(13,22,23,20)(15,18,24,17)(25,28,30,36)(26,32,29,38)(27,33,34,40)(31,39,37,35);(1,3,4,7)(2,6,5,8)(10,17)(13,23)(15,16)(18,19)(20,22)(21,24)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 216779);