#153352

order	512
nilpotency class	5
isoclinism	isoclinic class of #153081
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2218
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13476
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

206 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=de,b^2=dgh,d^2=ghi,e^2=gi,f^2=gi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fghi,g^a=gh,h^a=hi,c^b=cf,e^b=ei,d^c=dg,e^c=eghi,f^c=fghi,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320813, #72320842, #72320895
maximal normal subgroup of	#72320813, #72320842, #72320895

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60548);