#148228

order	512
nilpotency class	4
isoclinism	isoclinic class of #13442
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2054
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11076
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

663 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eh,g^c=gh,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, 55424);