#148215

order	512
nilpotency class	4
isoclinism	isoclinic class of #140874
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1982
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

673 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,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, 55411);