#149225

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	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

475 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=eh,d^2=h,e^2=i,f^2=i,g^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, 56421);