#498825

order	512
nilpotency class	4
isoclinism	isoclinic class of #13318
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1999
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1999
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1499 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)(6,26)(7,13)(8,14)(9,18)(10,23)(12,29)(16,47)(17,45)(19,46)(21,38)(22,40)(24,43)(25,42)(27,33)(28,37)(30,41)(31,39)(32,57)(34,61)(35,53)(36,60)(44,50)(48,63)(49,54)(51,64)(52,62)(55,58)(56,59);(1,3)(2,9)(4,16)(5,17)(6,27)(7,31)(8,19)(10,36)(11,33)(12,39)(13,34)(14,35)(15,26)(18,29)(20,54)(21,48)(22,57)(23,50)(24,51)(25,53)(28,60)(30,61)(32,38)(37,64)(40,47)(41,46)(42,45)(43,62)(44,55)(49,58)(52,56)(59,63);(1,4)(2,10)(3,16)(5,21)(6,28)(7,23)(8,24)(9,32)(11,20)(12,40)(13,43)(14,38)(15,44)(17,48)(18,52)(19,51)(22,30)(25,58)(26,59)(27,60)(29,56)(31,50)(33,62)(34,54)(35,36)(37,42)(39,64)(41,55)(45,47)(46,63)(49,53)(57,61);(1,5,8,7)(2,11,14,13)(3,17,19,31)(4,21,24,23)(6,25,30,29)(9,33,35,34)(10,20,38,43)(12,26,42,41)(15,45,46,39)(16,48,51,50)(18,27,53,61)(22,56,28,58)(32,62,36,54)(37,55,40,59)(44,47,63,64)(49,57,52,60)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249764, #72249772, #72249832, …
maximal normal subgroup of	#72249764, #72249772, #72249832, …
maximal quotient of	#72250201, C₂×#498825, #72322966

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406021);