#375674

order	512
nilpotency class	3
isoclinism	isoclinic class of #213728
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1120 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1120 conjugacy classes of subgroups — too many to draw. Only its 250 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)(3,7)(4,5)(6,8)(9,10,15,20,14,19,24,21)(11,16,23,12,17,27,30,26)(13,18,29,35,28,34,31,36)(22,32,37,39,33,40,38,25);(2,5)(6,7)(9,11,14,17)(10,16,19,27)(12,20,26,21)(13,22,28,33)(15,30,24,23)(18,32,34,40)(25,36,39,35)(29,38,31,37);(9,12,24,27,14,26,15,16)(10,17,20,23,19,11,21,30)(13,25,29,40,28,39,31,32)(18,33,36,38,34,22,35,37);(1,3,4,8)(2,6,5,7)(9,13,14,28)(10,18,19,34)(11,22,17,33)(12,25,26,39)(15,31,24,29)(16,32,27,40)(20,36,21,35)(23,38,30,37)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fhi,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282870);