#133504

order	512
nilpotency class	4
isoclinism	isoclinic class of #133362
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂²×Q₈	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

630 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 78 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)(11,21)(12,25,13,18)(14,19)(15,20)(16,22,33,27)(17,26,34,28)(23,37)(24,35)(29,31)(30,36)(32,38,40,39);(1,2)(3,5)(4,6)(7,8)(9,11,15,24)(10,16,20,33)(12,22,13,27)(14,29,30,23)(17,31,34,37)(18,35,25,21)(19,32,36,40)(26,39,28,38);(1,3)(2,5)(4,7)(6,8)(9,12,15,13)(10,17)(11,22,24,27)(14,26,30,28)(16,31)(18,36)(19,25)(20,34)(21,32)(23,38,29,39)(33,37)(35,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=di,e^b=ehi,f^b=fgi,d^c=dh,f^c=fi,e^d=ei,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, 40700);