#130436

order	512
nilpotency class	4
isoclinism	isoclinic class of #130252
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

565 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=i,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dg,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37632);