#505521

order	512
nilpotency class	4
isoclinism	isoclinic class of #505520
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#27534
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#578	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1765 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

527 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)(3,16)(4,10)(5,29)(6,15)(7,13)(8,14)(9,20)(11,31)(12,17)(18,49)(19,45)(21,47)(22,37)(23,57)(24,43)(25,48)(26,40)(27,41)(28,42)(30,59)(32,44)(33,60)(34,46)(35,51)(36,55)(38,54)(39,58)(50,63)(52,61)(53,64)(56,62);(1,3,17,22,7,21,6,20)(2,9,15,47,13,37,12,16)(4,23,25,58,28,50,27,53)(5,19,8,33,32,55,31,54)(10,39,41,57,42,64,48,63)(11,36,44,60,14,45,29,38)(18,24,51,26,52,34,56,30)(35,46,49,40,62,43,61,59);(1,4,6,25,7,28,17,27)(2,10,12,41,13,42,15,48)(3,18,20,51,21,52,22,56)(5,30,31,24,32,26,8,34)(9,35,16,49,37,62,47,61)(11,43,29,59,14,46,44,40)(19,53,54,23,55,58,33,50)(36,39,38,57,45,64,60,63);(1,5,6,31,7,32,17,8)(2,11,12,29,13,14,15,44)(3,19,20,54,21,55,22,33)(4,24,27,30,28,34,25,26)(9,36,16,38,37,45,47,60)(10,40,48,46,42,59,41,43)(18,50,56,58,52,23,51,53)(35,57,61,39,62,63,49,64)

Polycyclic presentation:

2^9|b^2=ef,c^2=e,d^2=efi,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gh,h^b=hi,d^c=dh,f^c=fi,g^c=ghi,h^c=hi,g^d=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412717);