#330943

order	512
nilpotency class	3
isoclinism	isoclinic class of #330923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

624 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 64

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,c^b=cg,e^b=eh,g^b=gi,d^c=de,f^c=fh,g^c=ghi,g^d=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238139);