#411975

order	512
nilpotency class	3
isoclinism	isoclinic class of #403107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2547
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15125
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1079 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)(3,9)(4,19,22,38)(5,24)(6,12)(7,29)(8,13)(10,23,37,28)(11,26)(14,40,42,35)(15,44)(16,32)(17,33)(18,34)(20,30,43,48)(21,36,51,60)(25,39)(27,54)(31,45)(41,62,63,56)(46,58)(47,57)(49,55,53,52)(50,61,64,59);(1,3)(2,9)(4,20,22,43)(5,15)(6,17)(7,16)(8,18)(10,35,37,40)(11,31)(12,33)(13,34)(14,23,42,28)(19,48,38,30)(21,50,51,64)(24,44)(25,46)(26,45)(27,47)(29,32)(36,59,60,61)(39,58)(41,52,63,55)(49,62,53,56)(54,57);(1,4)(2,10)(3,14)(5,21)(6,28)(7,23)(8,22)(9,30)(11,36)(12,38)(13,37)(15,41)(16,20)(17,43)(18,42)(19,29)(24,53)(25,55)(26,52)(27,51)(31,56)(32,35)(33,40)(34,48)(39,60)(44,61)(45,50)(46,64)(47,63)(49,54)(57,59)(58,62);(1,5,7,26,8,27,6,25)(2,11,12,24,13,39,29,54)(3,15,16,45,18,47,17,46)(4,21,23,52,22,51,28,55)(9,31,33,44,34,58,32,57)(10,36,38,53,37,60,19,49)(14,41,20,50,42,63,43,64)(30,56,40,61,48,62,35,59)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319171);