#496449

order	512
nilpotency class	4
isoclinism	isoclinic class of #496449
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21840
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

657 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)(3,14,18,8)(4,20,22,10)(5,24)(6,28)(7,11)(9,21,36,56)(12,26)(13,30)(15,46)(16,32)(17,52,53,57)(19,34,47,45)(23,39,29,41)(25,37,59,38)(27,55)(31,48)(33,54,44,63)(35,50)(40,60)(42,58)(43,51)(49,61)(62,64);(1,3,6,19,7,18,30,47)(2,8,28,45,11,14,13,34)(4,16,23,49,22,51,29,15)(5,17,26,54,27,53,60,63)(9,35,38,42,36,62,37,31)(10,32,41,61,20,43,39,46)(12,33,55,52,40,44,24,57)(21,50,25,58,56,64,59,48);(1,4)(2,9,11,36)(3,15)(5,25)(6,29)(7,22)(8,31,14,42)(10,24,20,55)(12,41,40,39)(13,38,28,37)(16,47)(17,50)(18,49)(19,51)(21,26)(23,30)(27,59)(32,44,43,33)(34,35,45,62)(46,57,61,52)(48,54)(53,64)(56,60)(58,63);(1,4)(2,10)(3,16)(5,21)(6,23)(7,22)(8,32)(9,24)(11,20)(12,38)(13,39)(14,43)(15,47)(17,50)(18,51)(19,49)(25,26)(27,56)(28,41)(29,30)(31,44)(33,42)(34,46)(35,57)(36,55)(37,40)(45,61)(48,63)(52,62)(53,64)(54,58)(59,60)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403645);