#148135

order	512
nilpotency class	4
isoclinism	isoclinic class of #13318
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#11066
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,e^c=eh,f^c=fh,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, 55331);