#144226

order	512
nilpotency class	4
isoclinism	isoclinic class of #144225
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10320
commutator, Frattini, cosocle	C₂×#131	C₂³
prime radical	#32951	C₂
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

556 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^e=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, 51422);