#142261

order	512
nilpotency class	4
isoclinism	isoclinic class of #142260
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	#23414	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

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

normal subgroups only

This group has 726 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)(3,15,19,45)(4,20,23,42)(5,24,28,40)(6,13,7,12)(9,32,35,17)(10,30,37,22)(11,27,41,29)(16,46,50,63)(18,51,31,38)(21,39,26,36)(25,43,57,34)(33,54,62,58)(44,49,64,48)(47,52,60,61)(53,59,56,55);(1,3)(2,9)(4,16)(5,25)(6,18)(7,31)(8,19)(10,33)(11,38)(12,34)(13,43)(14,35)(15,24)(17,29)(20,52)(21,54)(22,48)(23,50)(26,58)(27,32)(28,57)(30,49)(36,63)(37,62)(39,46)(40,45)(41,51)(42,61)(44,53)(47,55)(56,64)(59,60);(1,4)(2,10)(3,16)(5,26)(6,30)(7,22)(8,23)(9,33)(11,39)(12,42)(13,20)(14,37)(15,44)(17,47)(18,49)(19,50)(21,28)(24,56)(25,54)(27,55)(29,59)(31,48)(32,60)(34,61)(35,62)(36,41)(38,63)(40,53)(43,52)(45,64)(46,51)(57,58)

Polycyclic presentation:

2^9|a^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,g^b=gi,h^b=hi,d^c=dh,e^c=ei,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, 49457);