#148873

order	512
nilpotency class	4
isoclinism	isoclinic class of #13469
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2065
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11228
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

473 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,d^2=i,e^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eh,g^c=gh,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, 56069);