#152372

order	512
nilpotency class	5
isoclinism	isoclinic class of #152371
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12866
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#41	D₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

251 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224991, #72225001, #72225002, …
maximal normal subgroup of	#72224991, #72225001, #72225002, …
maximal quotient of	C₂×#152372

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59568);