#331456

order	512
nilpotency class	3
isoclinism	isoclinic class of #331233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

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

680 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 158 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)(11,14)(12,16)(13,15)(17,21)(18,22)(19,24)(20,23)(25,26,29,33,30,32,34,27)(28,35,37,36,38,40,39,31);(2,5)(6,8)(9,11,17,18,13,20,12,19)(10,14,21,22,15,23,16,24)(25,27,34,32,30,33,29,26)(28,36,39,35,38,31,37,40);(3,7)(6,8)(11,18)(12,17)(14,22)(16,21)(19,20)(23,24)(25,28,30,38)(26,31,32,36)(27,35,33,40)(29,39,34,37);(1,3,4,7)(2,6,5,8)(10,15)(14,23)(16,21)(22,24)(26,32)(27,33)(31,36)(35,40)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,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, 238652);