#134090

order	512
nilpotency class	4
isoclinism	isoclinic class of #133899
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂×#534	C₄
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

941 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 941 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ei,f^b=fg,d^c=di,e^c=ei,f^c=fh,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41286);