#153218

order	512
nilpotency class	5
isoclinism	isoclinic class of #153218
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13363
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

242 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,13,15,42)(4,19)(5,25)(6,30)(7,10)(8,16,14,46)(9,22)(11,28)(12,27)(17,53,48,64)(18,54,31,45)(20,39,51,43)(21,38)(23,61)(24,41)(26,44,50,34)(29,35)(32,62,47,57)(33,63,40,52)(36,49)(37,58)(55,60)(56,59);(1,2)(3,14,15,8)(4,9)(5,11)(6,12)(7,10)(13,16,42,46)(17,47,48,32)(18,40,31,33)(19,22)(20,44,51,34)(21,35)(23,37)(24,36)(25,28)(26,43,50,39)(27,30)(29,38)(41,49)(45,52,54,63)(53,62,64,57)(55,59)(56,60)(58,61);(1,3)(2,8)(4,20)(5,26)(6,31)(7,15)(9,34)(10,14)(11,39)(12,40)(13,41)(16,49)(17,28)(18,27)(19,42)(21,48)(22,46)(23,62)(24,51)(25,32)(29,50)(30,33)(35,47)(36,44)(37,53)(38,43)(45,56)(52,60)(54,61)(55,64)(57,59)(58,63)

Polycyclic presentation:

2^9|a^2=d,b^2=d,e^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,f^b=fi,d^c=di,e^c=egi,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224988, #72224999, #72225008, …
maximal normal subgroup of	#72224988, #72224999, #72225008, …
maximal quotient of	C₂×#153218

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60414);