#484571

order	512
nilpotency class	4
isoclinism	isoclinic class of #484570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#60773	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

801 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 801 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,e^2=g,g^2=i,b^a=be,c^a=cf,e^a=eg,g^a=gi,h^a=hi,e^b=egi,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391767);