#280919

order	512
nilpotency class	3
isoclinism	isoclinic class of #280915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

470 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 188115);