#416371

order	512
nilpotency class	3
isoclinism	isoclinic class of #396100
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2414
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15222
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

967 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=e,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,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, 323567);