#149227

order	512
nilpotency class	4
isoclinism	isoclinic class of #135850
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1854
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)(3,15,19,46,20,47,18,45)(4,21,25,10,26,39,24,40)(5,27,29,43,31,42,30,11)(9,32,35,17,36,54,34,55)(16,48,51,62,52,64,22,44)(23,41,59,57,28,38,49,60)(33,50,37,61,56,53,63,58);(1,3,5,17,7,19,30,54,8,20,31,55,6,18,29,32)(2,9,11,15,12,34,27,45,14,36,43,47,13,35,42,46)(4,22,23,58,25,52,59,53,26,51,28,61,24,16,49,50)(10,37,38,64,21,56,57,44,40,63,41,48,39,33,60,62);(1,4)(2,10,14,40)(3,16)(5,28)(6,24)(7,25)(8,26)(9,33,36,56)(11,41,43,38)(12,21,13,39)(15,44,47,62)(17,53)(18,51)(19,22)(20,52)(23,31)(27,60,42,57)(29,59)(30,49)(32,58)(34,37,35,63)(45,48,46,64)(50,55)(54,61)

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,g^b=gi,d^c=di,f^c=fi,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, 56423);