#153257

order	512
nilpotency class	5
isoclinism	isoclinic class of #153257
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13381
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	#33611	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 266 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)(3,14)(4,21)(5,11)(6,31)(7,29)(8,13)(9,17)(10,26)(12,24)(15,46)(16,43)(18,44)(19,53)(20,49)(22,54)(23,59)(25,55)(27,42)(28,57)(30,40)(32,45)(33,58)(34,47)(35,60)(36,52)(37,50)(38,41)(39,51)(48,64)(56,62)(61,63);(1,2)(3,15)(4,22)(5,29)(6,12)(7,11)(8,13)(9,18)(10,28)(14,41)(16,38)(17,49)(19,39)(20,34)(21,54)(23,53)(24,31)(25,37)(26,57)(27,56)(30,58)(32,63)(33,40)(35,52)(36,61)(42,62)(43,46)(44,47)(45,60)(48,59)(50,55)(51,64);(1,3,7,18,8,16,5,20)(2,9,11,38,13,34,29,15)(4,23,27,60,28,51,25,61)(6,32,24,19,33,52,30,48)(10,36,37,53,22,45,56,64)(12,39,31,35,40,59,58,63)(14,42,49,21,43,55,44,57)(17,50,46,26,47,62,41,54)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224988, #72225004, #72225005, …
maximal normal subgroup of	#72224988, #72225004, #72225005, …
maximal quotient of	C₂×#153257

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60453);