#146892

order	512
nilpotency class	4
isoclinism	isoclinic class of #12873
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1823
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10806
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

487 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)(3,14,17,9)(4,20,22,54)(5,24)(6,13)(7,29)(8,11)(10,21,34,56)(12,27)(15,42)(16,46,48,57)(18,41,19,33)(23,55,30,38)(25,36,58,35)(26,37)(28,53)(31,43)(32,50,40,49)(39,62)(44,61)(45,52)(47,59)(51,63)(60,64);(1,3,6,18,8,17,7,19)(2,9,13,33,11,14,29,41)(4,15,23,44,22,45,30,51)(5,16,26,49,28,48,27,50)(10,31,35,59,34,39,36,60)(12,32,24,57,37,40,53,46)(20,52,55,63,54,42,38,61)(21,43,25,47,56,62,58,64);(1,4)(2,10,11,34)(3,15)(5,25)(6,23)(7,30)(8,22)(9,31,14,39)(12,38,37,55)(13,35,29,36)(16,47)(17,45)(18,44)(19,51)(20,53,54,24)(21,27)(26,56)(28,58)(32,61,40,63)(33,59,41,60)(42,57,52,46)(43,50)(48,64)(49,62)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54088);