#130255

order	512
nilpotency class	4
isoclinism	isoclinic class of #130254
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#62279	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

600 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 92 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 — 3 generators on 40 points:

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

Polycyclic presentation:

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

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, 37451);