#153111

order	512
nilpotency class	5
isoclinism	isoclinic class of #153111
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13325
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

378 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,16)(4,9)(5,26)(6,30)(7,34)(8,13)(10,17)(11,21)(12,27)(14,23)(15,32)(18,53)(19,55)(20,36)(22,57)(24,45)(25,48)(28,47)(29,38)(31,46)(33,43)(35,50)(37,56)(39,52)(40,60)(41,61)(42,62)(44,59)(49,51)(54,63)(58,64);(2,9)(4,21)(5,27)(6,23)(7,32)(10,16)(11,14)(12,45)(13,26)(15,48)(17,28)(18,31)(19,35)(22,52)(24,33)(25,59)(29,56)(30,37)(34,38)(36,39)(40,53)(41,55)(42,57)(43,44)(46,47)(49,50)(51,58)(54,62)(60,61)(63,64);(1,3)(2,10)(4,22)(5,28)(6,31)(7,35)(8,20)(9,36)(11,42)(12,46)(13,39)(14,47)(15,49)(16,26)(17,27)(18,23)(19,32)(21,52)(24,54)(25,60)(29,58)(30,40)(33,62)(34,41)(37,63)(38,50)(43,64)(44,53)(45,57)(48,55)(51,56)(59,61)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224988, #72224993, #72224994, …
maximal normal subgroup of	#72224988, #72224993, #72224994, …
maximal quotient of	C₂×#153111

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60307);