#330363

order	512
nilpotency class	3
isoclinism	isoclinic class of #329743
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 584 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 237559);