#133990

order	512
nilpotency class	4
isoclinism	isoclinic class of #133918
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2297 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

721 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 721 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,7,4,6)(5,8)(9,10)(11,20)(12,17,25,34)(13,16)(14,19)(15,21,30,24)(18,28)(22,36)(23,32)(26,31,40,33)(27,37)(29,39,35,38);(1,3,5,4)(2,6,8,7)(9,11,14,23)(10,15,19,30)(12,24,25,21)(13,27,28,22)(16,29,18,35)(17,32,34,20)(26,38,40,39)(31,36,33,37);(1,4,5,3)(2,7,8,6)(10,16,19,18)(12,25)(15,29,30,35)(17,33,34,31)(20,36,32,37)(21,24)(26,40)(38,39)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gi,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=di,e^b=ei,f^b=fgi,d^c=di,f^c=fhi,f^d=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, 41186);