#331426

order	512
nilpotency class	3
isoclinism	isoclinic class of #331190
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

690 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 170 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)(12,15)(13,16)(17,21)(18,24)(19,23)(20,22)(25,26,30,34,31,28,29,33)(27,32,38,35,36,39,37,40);(2,5)(6,8)(9,11,17,18,12,19,13,20)(10,14,21,24,15,23,16,22)(25,27,29,37,31,36,30,38)(26,32,33,40,28,39,34,35);(3,7)(6,8)(10,15)(11,18)(13,17)(14,22)(19,20)(23,24)(25,28,31,26)(27,35,36,40)(29,33,30,34)(32,37,39,38);(1,3,4,7)(2,6,5,8)(27,36)(32,39)(35,40)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238622);