#137448

order	512
nilpotency class	4
isoclinism	isoclinic class of #12307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1616
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#9045
prime radical	#3054	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 364 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=g,e^2=g,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eghi,h^a=hi,d^b=dg,e^b=ehi,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fhi,g^c=gi,h^c=hi,e^d=ei,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44644);