#146906

order	512
nilpotency class	4
isoclinism	isoclinic class of #12826
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1804
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10807
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

483 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=dh,d^2=i,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,h^a=hi,e^b=ei,f^b=fi,d^c=di,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, 54102);