#498897

order	512
nilpotency class	4
isoclinism	isoclinic class of #13350
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2021
commutator, Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂×#2021
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,h^b=hi,e^c=eh,f^c=fi,g^e=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280488, #72281882, #72282105, …
maximal normal subgroup of	#72280488, #72281882, #72282105, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406093);