#149231

order	512
nilpotency class	4
isoclinism	isoclinic class of #135755
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1846
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:

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

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,h^a=hi,c^b=cf,e^b=ei,g^b=gi,d^c=di,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, 56427);