#147435

order	512
nilpotency class	4
isoclinism	isoclinic class of #13247
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1972
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

680 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)(5,6,10,16,12,18,33,17)(7,19,23,51,25,52,22,50)(8,26,29,46,31,48,30,14)(9,32,34,15,11,36,35,47)(13,38,41,24,42,58,40,21)(20,53,56,49,57,68,27,63)(28,45,54,62,37,44,65,61)(39,59,43,64,60,55,67,66);(2,4)(5,7,11,24,33,22,34,38,12,25,9,21,10,23,35,58)(6,13,15,19,16,41,36,51,18,42,47,52,17,40,32,50)(8,27,28,64,29,20,54,55,31,56,37,66,30,57,65,59)(14,43,45,53,48,39,61,63,46,67,44,68,26,60,62,49);(5,8,12,31)(6,14,18,46)(7,20)(9,28,11,37)(10,30,33,29)(13,39)(15,44,47,45)(16,48,17,26)(19,49)(21,55)(22,56)(23,27)(24,59)(25,57)(32,61,36,62)(34,65,35,54)(38,64)(40,43)(41,67)(42,60)(50,68)(51,53)(52,63)(58,66)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,c^b=ch,e^b=ei,e^c=ei,f^c=fi,g^c=gi

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, 54631);