#318764

order	512
nilpotency class	3
isoclinism	isoclinic class of #262851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

514 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 158 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)(9,10,12,15)(11,17,19,24)(13,22,21,16)(14,23,18,20)(25,26)(27,31)(28,32)(29,33)(30,34)(35,40)(36,38)(37,39);(1,3,5,4)(2,6,8,7)(9,10)(11,18)(12,15)(13,16)(14,19)(17,23)(20,24)(21,22)(25,27,28,35,30,36,29,37)(26,31,32,40,34,38,33,39);(9,11,12,19)(10,14,15,18)(13,23,21,20)(16,24,22,17)(26,32)(27,35)(28,29)(31,38)(33,34)(36,37);(1,4,5,3)(2,7,8,6)(10,15)(14,18)(16,22)(17,24)(27,36)(31,38)(35,37)(39,40)

Polycyclic presentation:

2^9|a^2=e,b^2=egi,c^2=hi,d^2=ehi,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, 225960);