#149004

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	#11268
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

431 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)(3,16)(4,24,29,57)(5,31)(6,11)(7,13)(8,15)(9,19)(10,27,41,60)(12,17)(14,32)(18,50,53,34)(20,43)(21,46)(22,35)(23,47)(25,58,55,38)(26,44,49,40)(28,39,30,61)(33,42)(36,54)(37,48)(45,52,56,64)(51,62,59,63);(1,3,17,48,32,21,6,20,8,23,5,19,7,22,33,54)(2,9,11,16,13,43,12,35,15,37,42,47,14,36,31,46)(4,25,49,64,28,18,27,59,29,55,26,52,30,53,60,51)(10,38,39,62,44,50,24,56,41,58,61,63,40,34,57,45);(1,4,5,26,32,28,33,60,8,29,17,49,7,30,6,27)(2,10,11,39,13,44,12,24,15,41,42,61,14,40,31,57)(3,18,19,51,21,55,54,64,23,53,48,59,22,25,20,52)(9,34,16,45,43,38,35,62,37,50,47,56,36,58,46,63)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,f^b=fi,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, 56200);