#144241

order	512
nilpotency class	4
isoclinism	isoclinic class of #144240
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

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

normal subgroups only

This group has 560 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=gi,e^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,g^b=gi,e^c=ei,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, 51437);