#144636

order	512
nilpotency class	4
isoclinism	isoclinic class of #144636
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10390
commutator, Frattini, cosocle	C₂²×#41	C₂³
prime radical	#3487	C₂²
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

434 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 46 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)(3,8)(4,17,19,37)(5,22,24,10)(6,11)(7,12)(9,20)(13,43,44,61)(14,36,47,30)(15,39)(16,32)(18,54,49,52)(21,58,26,53)(23,35)(25,41,27,38)(28,42)(29,45)(31,51)(33,56)(34,59)(40,57)(46,60)(48,62,50,64)(55,63);(1,3)(2,8)(4,18,19,49)(5,14,24,47)(6,15)(7,16)(9,33)(10,36,22,30)(11,39)(12,32)(13,21,44,26)(17,52,37,54)(20,56)(23,55)(25,48,27,50)(28,51)(29,34)(31,42)(35,63)(38,62,41,64)(40,60)(43,53,61,58)(45,59)(46,57);(1,4)(2,9)(3,13)(5,23)(6,26)(7,19)(8,29)(10,37)(11,40)(12,35)(14,46)(15,49)(16,44)(17,22)(18,51)(20,24)(21,28)(25,57)(27,59)(30,61)(31,63)(32,60)(33,39)(34,42)(36,43)(38,53)(41,58)(45,47)(48,56)(50,55)(52,64)(54,62)

Polycyclic presentation:

2^9|a^2=d,b^2=d,e^2=i,f^2=i,b^a=bd,c^a=ce,e^a=egi,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fgi,h^b=hi,d^c=dg,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^e=fi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72268778, #72268780

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51832);