#512611

order	512
nilpotency class	5
isoclinism	isoclinic class of #512609
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#353
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#124	C₂⁴
prime radical	#60651	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×D₄ ◁ C₂×#41 ◁ #3500 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1080 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)(3,13,16,37)(4,9)(5,21,22,51)(6,24,26,11)(7,12)(8,15,28,42)(10,23)(14,36,41,58)(17,34,25,38)(18,31)(19,47,45,63)(20,49,46,33)(27,40,54,61)(29,43,30,44)(32,48)(35,52)(39,57,50,59)(53,60,55,62)(56,64);(1,3,7,17)(2,8,12,30)(4,14,18,39)(5,15,22,44)(6,25,26,16)(9,27,31,53)(10,34,35,13)(11,28,24,29)(19,40,45,60)(20,50,46,41)(21,38,51,37)(23,42,52,43)(32,57,56,36)(33,54,49,55)(47,59,63,58)(48,61,64,62);(1,4)(2,9)(3,14)(5,19)(6,20)(7,18)(8,27)(10,32)(11,33)(12,31)(13,36)(15,40)(16,41)(17,39)(21,47)(22,45)(23,48)(24,49)(25,50)(26,46)(28,54)(29,55)(30,53)(34,57)(35,56)(37,58)(38,59)(42,61)(43,62)(44,60)(51,63)(52,64);(4,18)(9,31)(14,39)(19,45)(20,46)(27,53)(32,56)(33,49)(36,57)(40,60)(41,50)(47,63)(48,64)(54,55)(58,59)(61,62)

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=h,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,d^c=di,f^e=fhi,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419807);