#396791

order	512
nilpotency class	3
isoclinism	isoclinic class of #394483
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2472
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14738
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

947 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:

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

Polycyclic presentation:

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

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, 303987);