#7622588

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	#3584	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1885 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224948, #72224961, #72224962, …
maximal normal subgroup of	#72224948, #72224961, #72224962, …
maximal quotient of	#72286610, D₄×#2171, C₂×#7622588, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529784);