#7622586

order	512
nilpotency class	3
isoclinism	isoclinic class of #3585
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂³×D₄
commutator	C₄	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂	C₂²×C₈×D₄
prime radical	C₂×#575	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1897 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,f^2=i,g^2=h,h^2=i,b^a=bf,e^a=ei,f^a=fi,f^b=fi,d^c=di

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285581, #72285601, #72285648, …
maximal normal subgroup of	#72285581, #72285601, #72285648, …
maximal quotient of	M₃₂×AGL(1,ℤ/8), #72321224

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529782);