#148774

order	512
nilpotency class	4
isoclinism	isoclinic class of #13391
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2033
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11219
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

463 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,e^c=ehi,f^c=fhi,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, 55970);