#133778

order	512
nilpotency class	4
isoclinism	isoclinic class of #133374
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂×#538	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#542 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 636 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) = 64
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,e^2=i,f^2=hi,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, 40974);