#146878

order	512
nilpotency class	4
isoclinism	isoclinic class of #128583
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10806
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ 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)(3,14,18,41)(4,20)(5,11)(6,25)(7,29)(8,13)(9,16,33,19)(10,22)(12,27)(15,42,45,60)(17,47,49,32)(21,53)(23,54)(24,38)(26,34)(28,56)(30,37)(31,48,58,64)(35,55)(36,57)(39,61,59,44)(40,46,62,50)(43,63,51,52);(1,3)(2,9,13,33)(4,15)(5,16)(6,17)(7,19)(8,18)(10,31,35,58)(11,14,29,41)(12,32,38,47)(20,52,54,63)(21,43)(22,44)(23,45)(24,46)(25,40,56,62)(26,48)(27,50)(28,49)(30,51)(34,39,36,59)(37,60,53,42)(55,61)(57,64);(1,4,5,21,8,23,7,30)(2,10,11,34,13,35,29,36)(3,15,16,43,18,45,19,51)(6,26,24,55,28,57,27,22)(9,31,14,39,33,58,41,59)(12,37,25,20,38,53,56,54)(17,48,46,61,49,64,50,44)(32,60,40,52,47,42,62,63)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,h^a=hi,f^b=fi,e^c=eh,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, 54074);