#142263

order	512
nilpotency class	4
isoclinism	isoclinic class of #142262
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9914
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#23434	C₂
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

710 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

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,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49459);