#149268

order	512
nilpotency class	4
isoclinism	isoclinic class of #13318
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11328
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

505 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=e,d^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ei,d^c=dhi,f^c=fhi,h^c=hi

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, 56464);