#148381

order	512
nilpotency class	4
isoclinism	isoclinic class of #140781
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1972
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

660 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)(5,6,10,17,13,19,12,40)(7,20,25,56,26,44,23,55)(8,27,30,50,33,52,32,15)(9,34,36,18,11,38,37,16)(14,41,45,22,46,63,43,24)(21,57,60,54,61,47,59,68)(28,42,62,66,58,67,64,65)(29,49,35,51,31,48,39,53);(1,2,3,4)(5,7,9,22,12,25,37,63,13,26,11,24,10,23,36,41)(6,14,16,20,17,43,34,55,19,46,18,44,40,45,38,56)(8,28,29,59,32,62,39,21,33,58,31,60,30,64,35,61)(15,47,48,66,27,54,53,42,50,57,49,65,52,68,51,67);(5,8,13,33)(6,15,19,50)(7,21,26,61)(9,35,11,39)(10,32,12,30)(14,42,46,67)(16,51,18,53)(17,52,40,27)(20,54,44,68)(22,62,24,64)(23,60,25,59)(28,41,58,63)(29,37,31,36)(34,49,38,48)(43,66,45,65)(47,56,57,55)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,g^b=gi,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, 55577);