#148230

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, socle	C₂²	#2054
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

666 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,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,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, 55426);