#512162

order	512
nilpotency class	4
isoclinism	isoclinic class of #511841
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3027
commutator, Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	#33021
prime radical	#23995	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

769 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)(4,18)(5,22)(6,11)(7,12)(8,16)(9,20)(10,24)(14,38)(15,36)(17,37)(19,49)(21,33)(23,47)(25,31)(26,34)(27,55)(28,42)(29,44)(30,54)(32,52)(35,41)(39,61)(40,46)(43,60)(45,59)(48,51)(50,53)(56,64)(57,63)(58,62);(1,3)(2,8,12,28)(4,14)(5,16)(6,17)(7,15)(9,27,31,56)(10,13,22,36)(11,29,34,57)(18,46,33,58)(19,40)(20,41)(21,39)(23,44)(24,42)(25,45)(26,43)(30,35,53,59)(32,37,47,60)(38,49,61,48)(50,55)(51,62)(52,63)(54,64);(1,4,5,19,7,21,24,51)(2,9,10,30,12,31,22,53)(3,14,16,40,15,39,42,62)(6,25,23,54,26,20,52,50)(8,27,13,35,28,56,36,59)(11,33,32,49,34,18,47,48)(17,45,44,64,43,41,63,55)(29,58,37,61,57,46,60,38);(3,15)(8,28)(11,34)(13,36)(14,39)(16,42)(17,43)(18,33)(27,56)(32,47)(35,59)(40,62)(41,45)(44,63)(48,49)(55,64)

Polycyclic presentation:

2^9|b^2=h,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^a=hi,d^b=di,f^c=fi,g^c=ghi,f^d=fh,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, 419358);