#498837

order	512
nilpotency class	4
isoclinism	isoclinic class of #13323
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

1481 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)(6,28)(7,13)(8,14)(9,18)(10,25)(12,31)(16,48)(17,46)(19,40)(20,47)(22,56)(23,39)(24,41)(26,44)(27,43)(29,34)(30,38)(32,42)(33,57)(35,53)(36,54)(37,49)(45,50)(51,64)(52,62)(55,59)(58,60)(61,63);(1,3)(2,9)(4,22)(5,17)(6,29)(7,19)(8,20)(10,37)(11,34)(12,40)(13,35)(14,36)(15,28)(16,25)(18,31)(21,33)(23,51)(24,52)(26,50)(27,54)(30,49)(32,53)(38,45)(39,62)(41,56)(42,47)(43,46)(44,63)(48,55)(57,59)(58,61)(60,64);(1,4)(2,10)(3,16)(5,23)(6,30)(7,25)(8,26)(9,33)(11,21)(12,41)(13,44)(14,39)(15,45)(17,22)(18,52)(19,50)(20,51)(24,32)(27,59)(28,60)(29,61)(31,58)(34,62)(35,37)(36,63)(38,43)(40,64)(42,55)(46,48)(47,56)(49,54)(53,57);(1,5,8,7)(2,11,14,13)(3,17,20,19)(4,23,26,25)(6,27,32,31)(9,34,36,35)(10,21,39,44)(12,28,43,42)(15,46,47,40)(16,22,51,50)(18,29,54,53)(24,58,30,59)(33,62,63,37)(38,55,41,60)(45,48,56,64)(49,57,52,61)

Polycyclic presentation:

2^9|d^2=i,e^2=hi,f^2=i,g^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,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320565, #72320602, #72320688
maximal normal subgroup of	#72320565, #72320602, #72320688
maximal quotient of	#72322930

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406033);