#149224

order	512
nilpotency class	4
isoclinism	isoclinic class of #140408
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1949
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11325
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

467 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=eh,d^2=h,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,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, 56420);