#6371341

order	512
nilpotency class	3
isoclinism	isoclinic class of #60399
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#120
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂²×#385
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4465 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317340, #72317343, #72317348, …
maximal normal subgroup of	#72317340, #72317343, #72317348, …
maximal quotient of	D₄×#1868

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278537);