#316867

order	512
nilpotency class	3
isoclinism	isoclinic class of #262851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

566 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 146 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)(3,7)(4,5)(6,8)(9,10,12,17)(11,19,21,15)(13,16,14,18)(20,23,22,24);(2,5)(6,7)(9,11,14,22,12,21,13,20)(10,15,18,23,17,19,16,24)(25,26,29,33,30,34,28,32)(27,35,36,39,37,40,38,31);(10,16)(11,20)(13,14)(15,19)(17,18)(21,22)(25,27,30,37)(26,31,34,39)(28,36,29,38)(32,35,33,40);(1,3,4,8)(2,6,5,7)(10,17)(11,21)(16,18)(20,22)(25,28,30,29)(26,32,34,33)(27,36,37,38)(31,35,39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,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, 224063);