#147162

order	512
nilpotency class	3
isoclinism	isoclinic class of #11115
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#388
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10877
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

491 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54358);