#133381

order	512
nilpotency class	4
isoclinism	isoclinic class of #133374
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

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

normal subgroups only

This group has 814 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,e^b=eh,f^b=fg,d^c=dh,e^c=ei,f^c=fi,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, 40577);