#147158

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

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,g^a=gi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54354);