#498872

order	512
nilpotency class	4
isoclinism	isoclinic class of #13334
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2011
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2011
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1539 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)(4,19)(5,11)(6,25)(7,30)(8,13)(9,17)(10,22)(12,28)(15,33)(16,46)(18,47)(20,56)(21,40)(23,39)(24,44)(26,34)(27,38)(29,59)(31,42)(32,43)(35,54)(36,60)(37,52)(41,57)(45,50)(48,61)(49,55)(51,64)(53,63)(58,62);(1,3)(2,9)(4,15)(5,16)(6,26)(7,31)(8,18)(10,37)(11,34)(12,42)(13,35)(14,25)(17,28)(19,55)(20,48)(21,58)(22,53)(23,52)(24,54)(27,50)(29,60)(30,36)(32,51)(33,40)(38,64)(39,45)(41,61)(43,62)(44,46)(47,59)(49,57)(56,63);(1,4)(2,10)(3,15)(5,20)(6,27)(7,32)(8,23)(9,33)(11,38)(12,43)(13,40)(14,45)(16,48)(17,53)(18,52)(19,25)(21,24)(22,28)(26,50)(29,57)(30,41)(31,51)(34,61)(35,37)(36,64)(39,59)(42,63)(44,56)(46,62)(47,55)(49,60)(54,58);(1,5,8,7)(2,11,13,30)(3,16,18,31)(4,20,23,32)(6,24,29,28)(9,34,35,36)(10,38,40,41)(12,25,44,59)(14,46,47,42)(15,48,52,51)(17,26,54,60)(19,56,39,43)(21,57,22,27)(33,61,37,64)(45,62,55,63)(49,53,50,58)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249769, #72249772, #72249830, …
maximal normal subgroup of	#72249769, #72249772, #72249830, …
maximal quotient of	#72250030, C₂×#498872, #72323013

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406068);