#331465

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	C₂³	C₂×#127
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

711 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 711 conjugacy classes of subgroups — too many to draw. Only its 154 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)(9,10,14,23,12,17,13,18)(11,19,22,16,20,15,21,24)(25,26,31,38,29,28,30,33)(27,34,39,35,36,32,37,40);(2,5)(6,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,18)(17,19)(23,24)(25,27,30,37,29,36,31,39)(26,32,33,35,28,34,38,40);(3,7)(6,8)(9,11)(10,16)(12,20)(13,22)(14,21)(15,18)(17,24)(19,23)(25,28,29,26)(27,35,36,40)(30,33,31,38)(32,39,34,37);(1,3,4,7)(2,6,5,8)(11,20)(15,19)(16,24)(21,22)(26,28)(27,36)(33,38)(37,39)

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=i,d^2=eh,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, 238661);