#146935

order	512
nilpotency class	4
isoclinism	isoclinic class of #129125
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1776
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10821
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

451 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:

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

Polycyclic presentation:

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

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, 54131);