#129572

order	512
nilpotency class	3
isoclinism	isoclinic class of #8534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1779
prime radical	#3459	C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

634 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 114 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)(9,10,12,17,14,18,25,32,13,16,24,29,26,30,37,39)(11,19,21,31,22,33,35,40,20,15,23,27,34,28,36,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,23)(13,20)(14,22)(16,19)(17,31)(18,28)(21,24)(25,36)(26,34)(27,29)(30,33)(32,40)(35,37)(38,39);(1,2)(3,5)(4,6)(7,8)(11,20)(15,19)(21,23)(22,34)(27,31)(28,33)(35,36)(38,40)

Polycyclic presentation:

2^9|a^2=f,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci,e^b=egi,f^b=fg,d^c=dgi

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, 36768);