#130737

order	512
nilpotency class	4
isoclinism	isoclinic class of #130736
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	#61057	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

811 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 811 conjugacy classes of subgroups — too many to draw. Only its 98 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)(9,10,15,21,17,20,13,32,16,22,35,40,14,36,34,19)(11,23,26,18,28,39,24,37,27,29,33,12,25,31,38,30);(1,3)(2,5)(4,8)(6,7)(9,11,16,27)(10,18,22,12)(13,33,34,26)(14,28,17,25)(15,38,35,24)(19,39,32,31)(20,30,36,37)(21,23,40,29);(1,3)(2,5)(4,8)(6,7)(9,12)(10,11)(13,23)(14,37)(15,31)(16,18)(17,30)(19,38)(20,28)(21,26)(22,27)(24,32)(25,36)(29,34)(33,40)(35,39)

Polycyclic presentation:

2^9|a^2=f,b^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=eg,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=ei,f^c=fgi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37933);