#148732

order	512
nilpotency class	4
isoclinism	isoclinic class of #13417
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2041
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11207
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

501 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,f^b=fi,e^c=ehi,f^c=fi,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, 55928);