#498817

order	512
nilpotency class	4
isoclinism	isoclinic class of #13310
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)(4,21)(5,11)(6,27)(7,13)(8,14)(9,18)(10,24)(12,29)(16,45)(17,43)(19,38)(20,44)(22,37)(23,55)(25,41)(26,40)(28,32)(30,39)(31,56)(33,50)(34,49)(35,60)(36,58)(42,46)(47,52)(48,62)(51,61)(53,57)(54,63)(59,64);(1,3)(2,9)(4,16)(5,17)(6,28)(7,19)(8,20)(10,35)(11,32)(12,38)(13,34)(14,33)(15,27)(18,29)(21,52)(22,46)(23,56)(24,51)(25,48)(26,50)(30,49)(31,37)(36,62)(39,44)(40,43)(41,59)(42,63)(45,55)(47,57)(53,61)(54,64)(58,60);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,31)(11,21)(12,36)(13,41)(14,37)(15,42)(17,46)(18,47)(19,51)(20,48)(26,54)(27,53)(28,56)(29,57)(30,58)(32,59)(33,35)(34,52)(38,45)(39,63)(40,55)(43,62)(44,61)(49,60)(50,64);(1,5,8,7)(2,11,14,13)(3,17,20,19)(4,22,25,24)(6,26,30,29)(9,32,33,34)(10,21,37,41)(12,27,40,39)(15,43,44,38)(16,46,48,51)(18,28,50,49)(23,54,58,57)(31,59,35,52)(36,53,55,63)(42,62,61,45)(47,56,64,60)

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,f^b=fi,h^b=hi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249757, #72249772, #72249836, …
maximal normal subgroup of	#72249757, #72249772, #72249836, …
maximal quotient of	#72249977, C₂×#498817

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406013);