#323531

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

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

630 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 186 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=egh,c^2=hi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230727);