#331493

order	512
nilpotency class	3
isoclinism	isoclinic class of #331276
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

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

normal subgroups only

This group has 703 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)(9,10,13,16,12,15,20,21)(11,17,19,24,18,14,22,23)(25,26,29,33,28,32,35,39)(27,34,37,40,36,30,38,31);(2,5)(6,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,22)(21,24)(25,27,35,38,28,36,29,37)(26,30,39,40,32,34,33,31);(3,7)(6,8)(9,10)(11,17)(12,15)(13,21)(14,18)(16,20)(19,23)(22,24)(25,27,28,36)(26,31,32,40)(29,38,35,37)(30,39,34,33);(1,3,4,7)(2,6,5,8)(10,15)(11,18)(16,21)(19,22)(27,36)(30,34)(31,40)(37,38)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=i,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,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, 238689);