#404597

order	512
nilpotency class	3
isoclinism	isoclinic class of #396612
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2850
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14894
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

751 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)(3,15)(4,21)(5,27)(6,13)(7,12)(8,14)(9,18)(10,25)(11,28)(16,42)(17,32)(19,33)(20,40)(22,31)(23,51)(24,36)(26,37)(29,38)(30,39)(34,45)(35,54)(41,47)(43,62)(44,58)(46,55)(48,59)(49,52)(50,61)(53,56)(57,64)(60,63);(1,3)(2,9)(4,22,25,44)(5,17)(6,19)(7,18)(8,20)(10,34,21,42)(11,32)(12,40)(13,15)(14,33)(16,24,45,26)(23,50,53,63)(27,55)(28,47)(29,48)(30,46)(31,37,58,36)(35,60,49,61)(38,41)(39,59)(43,52,64,54)(51,62,56,57);(1,4,6,24,8,25,7,26)(2,10,12,36,14,21,13,37)(3,16,18,44,20,45,19,22)(5,23,28,52,29,53,30,54)(9,31,15,34,33,58,40,42)(11,35,27,56,39,49,38,51)(17,43,46,63,48,64,47,50)(32,57,41,60,59,62,55,61);(1,5,6,28,8,29,7,30)(2,11,13,38,14,39,12,27)(3,17,19,47,20,48,18,46)(4,23,24,52,25,53,26,54)(9,32,15,41,33,59,40,55)(10,35,37,51,21,49,36,56)(16,43,22,50,45,64,44,63)(31,57,34,60,58,62,42,61)

Polycyclic presentation:

2^9|b^2=f,c^2=e,d^2=g,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^c=fi,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, 311793);