#498940

order	512
nilpotency class	4
isoclinism	isoclinic class of #13353
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

2094 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)(6,25)(7,30)(8,13)(9,17)(10,21)(12,29)(15,43)(16,42)(18,49)(19,39)(22,54)(23,35)(24,56)(26,34)(27,52)(28,38)(31,37)(32,59)(33,47)(36,57)(40,53)(41,44)(45,51)(46,63)(48,58)(50,62)(55,60)(61,64);(1,3)(2,9)(4,15)(5,16)(6,26)(7,18)(8,19)(10,32)(11,33)(12,39)(13,34)(14,25)(17,29)(20,51)(21,44)(22,55)(23,46)(24,48)(27,45)(28,47)(30,58)(31,50)(35,60)(36,62)(37,61)(38,49)(40,41)(42,56)(43,54)(52,63)(53,64)(57,59);(1,4)(2,10)(3,15)(5,21)(6,27)(7,31)(8,23)(9,32)(11,35)(12,40)(13,37)(14,41)(16,44)(17,45)(18,50)(19,46)(20,30)(22,29)(24,57)(25,36)(26,55)(28,53)(33,60)(34,61)(38,52)(39,62)(42,63)(43,49)(47,59)(48,64)(51,58)(54,56);(1,5,8,7)(2,11,13,30)(3,16,19,18)(4,21,23,31)(6,24,29,28)(9,33,34,58)(10,35,37,20)(12,38,25,56)(14,42,39,49)(15,44,46,50)(17,47,26,48)(22,53,27,57)(32,60,61,51)(36,54,40,52)(41,63,62,43)(45,59,55,64)

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=ghi,c^b=cg,e^b=ei,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	#72280539, #72281882, #72282105, …
maximal normal subgroup of	#72280539, #72281882, #72282105, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406136);