#133274

order	512
nilpotency class	4
isoclinism	isoclinic class of #133273
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8524
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄×M₁₆	C₂³
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

483 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 483 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,e^c=ei,f^c=fh,f^d=fi,f^e=fi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40470);