#147166

order	512
nilpotency class	3
isoclinism	isoclinic class of #11247
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#387
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10877
prime radical	C₂⁴	#131
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:

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

Polycyclic presentation:

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

Relationships

maximal subgroup of	#72257881, #72265417, #72265455
maximal normal subgroup of	#72257881, #72265417, #72265455

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54362);