#148438

order	512
nilpotency class	4
isoclinism	isoclinic class of #140140
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1925
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11123
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

669 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)(3,16)(4,22,27,45)(5,14)(6,13)(7,12)(8,34)(9,15)(10,18)(11,25,43,28)(17,51,54,62)(19,37)(20,38)(21,50)(23,59,55,63)(24,47,33,41)(26,42,60,58)(29,44)(30,48)(31,46)(32,36)(35,52,64,61)(39,56)(40,57,49,53);(1,3,7,32,8,19,30,56,9,21,29,20,6,31,5,18)(2,10,14,46,13,38,44,50,15,39,48,37,34,36,12,16)(4,23,60,53,28,61,33,54,27,55,26,57,25,52,24,17)(11,40,58,59,22,51,47,64,43,49,42,63,45,62,41,35);(1,4)(2,11)(3,17)(5,26)(6,28)(7,33)(8,25)(9,27)(10,35)(12,42)(13,45)(14,47)(15,43)(16,49)(18,55)(19,57)(20,52)(21,54)(22,34)(23,56)(24,29)(30,60)(31,53)(32,61)(36,59)(37,62)(38,63)(39,64)(40,50)(41,48)(44,58)(46,51)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,d^c=dhi,e^c=ei,f^c=fhi,g^c=gi,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, 55634);