#316832

order	512
nilpotency class	3
isoclinism	isoclinic class of #262851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 680 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=e,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, 224028);