#148422

order	512
nilpotency class	4
isoclinism	isoclinic class of #13472
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2067
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,4)(5,6,10,16,12,17,11,36)(7,18,24,51,25,40,22,50)(8,26,29,45,31,48,30,14)(9,32,34,47,35,65,19,15)(13,37,41,23,42,52,39,21)(20,53,56,49,57,43,55,60)(27,61,58,66,54,38,59,67)(28,62,63,68,64,44,33,46);(1,2,3,4)(5,7,19,52,11,22,35,23,12,25,34,37,10,24,9,21)(6,13,32,51,16,41,47,40,17,42,65,50,36,39,15,18)(8,27,33,57,30,59,64,56,31,54,63,20,29,58,28,55)(14,43,46,61,26,60,62,66,45,53,68,38,48,49,44,67);(5,8,12,31)(6,14,17,45)(7,20,25,57)(9,33,35,63)(10,30,11,29)(13,38,42,61)(15,46,47,68)(16,48,36,26)(18,49,40,60)(19,28,34,64)(21,58,23,59)(22,56,24,55)(27,52,54,37)(32,44,65,62)(39,67,41,66)(43,51,53,50)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=i,d^2=hi,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=ei,f^b=fi,d^c=dh,e^c=ei,f^c=fh,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, 55618);