#133812

order	512
nilpotency class	4
isoclinism	isoclinic class of #133422
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2312 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

484 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,11,6,12,23,38)(3,13,16,36,15,27,45,56)(4,9,20,31,18,33,49,60)(7,10,22,34,24,35,51,61)(8,25,28,46,29,52,58,17)(14,40,43,63,41,39,62,57)(19,32,21,37,48,30,50,59)(26,53,55,44,54,47,64,42);(1,3,7,17)(2,8,10,13)(4,14,19,44)(5,16,22,25)(6,15,24,46)(9,26,32,39)(11,36,34,28)(12,29,35,27)(18,41,48,42)(20,43,21,47)(23,45,51,52)(30,40,33,54)(31,57,37,55)(38,56,61,58)(49,62,50,53)(59,64,60,63);(1,4)(2,9,10,32)(3,14)(5,21)(6,18)(7,19)(8,26,13,39)(11,37,34,31)(12,33,35,30)(15,41)(16,47)(17,44)(20,22)(23,50)(24,48)(25,43)(27,40,29,54)(28,57,36,55)(38,59,61,60)(42,46)(45,53)(49,51)(52,62)(56,64,58,63)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,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, 41008);