#133901

order	512
nilpotency class	4
isoclinism	isoclinic class of #133883
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#537 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 911 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)(3,7,8,5)(4,6)(9,10)(11,20,23,32)(12,17,25,34)(13,16)(14,19)(15,21)(18,28)(22,36,27,37)(24,30)(26,31,40,33)(29,39)(35,38);(9,11)(10,15,19,30)(12,24)(13,27)(14,23)(16,29,18,35)(17,32,34,20)(21,25)(22,28)(26,38)(31,36,33,37)(39,40);(1,3)(2,5)(4,8)(6,7)(10,16)(12,25)(13,28)(15,29)(17,33)(18,19)(20,36)(21,24)(22,27)(30,35)(31,34)(32,37)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,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=fg,d^c=di,e^c=ei,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, 41097);