#498915

order	512
nilpotency class	4
isoclinism	isoclinic class of #13360
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

2076 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)(3,15)(4,20)(5,11)(6,27)(7,13)(8,14)(9,18)(10,24)(12,30)(16,48)(17,46)(19,41)(21,45)(22,39)(23,57)(25,44)(26,43)(28,36)(29,54)(31,47)(32,40)(33,55)(34,52)(35,53)(37,60)(38,56)(42,59)(49,64)(50,61)(51,63)(58,62);(1,3)(2,9)(4,21)(5,17)(6,28)(7,31)(8,19)(10,37)(11,34)(12,41)(13,35)(14,36)(15,27)(16,22)(18,30)(20,33)(23,58)(24,51)(25,49)(26,53)(29,55)(32,52)(38,48)(39,61)(40,47)(42,63)(43,46)(44,62)(45,57)(50,59)(54,64)(56,60);(1,4)(2,10)(3,16)(5,22)(6,29)(7,24)(8,25)(9,33)(11,20)(12,42)(13,44)(14,39)(15,45)(17,49)(18,50)(19,51)(21,31)(23,30)(26,59)(27,38)(28,60)(32,56)(34,61)(35,37)(36,62)(40,57)(41,64)(43,54)(46,48)(47,63)(52,58)(53,55);(1,5,8,7)(2,11,14,13)(3,17,19,31)(4,22,25,24)(6,26,30,32)(9,34,36,35)(10,20,39,44)(12,40,27,43)(15,46,41,47)(16,49,51,21)(18,52,28,53)(23,56,29,59)(33,61,62,37)(38,54,42,57)(45,48,64,63)(50,58,60,55)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303731, #72303737, #72303747, …
maximal normal subgroup of	#72303731, #72303737, #72303747, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406111);