#416390

order	512
nilpotency class	3
isoclinism	isoclinic class of #401790
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2485
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15223
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

881 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efh,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,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, 323586);