#331512

order	512
nilpotency class	3
isoclinism	isoclinic class of #331304
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

716 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 166 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)(9,10,13,16,12,14,20,23)(11,17,19,22,18,24,21,15)(26,30)(32,38)(33,39)(34,40);(2,5)(6,8)(10,14)(15,22)(16,23)(17,24)(25,26,31,39,28,30,29,33)(27,34,37,32,35,40,36,38);(3,7)(6,8)(9,11,12,18)(10,15,14,22)(13,21,20,19)(16,24,23,17)(25,27)(26,32)(28,35)(29,37)(30,38)(31,36)(33,40)(34,39);(1,3,4,7)(2,6,5,8)(10,14)(11,18)(16,23)(19,21)(26,30)(27,35)(33,39)(36,37)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=h,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,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, 238708);