#148907

order	512
nilpotency class	4
isoclinism	isoclinic class of #134701
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1809
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11232
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

465 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=hi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,h^a=hi,c^b=cf,e^b=ei,e^c=ei,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, 56103);