#149089

order	512
nilpotency class	4
isoclinism	isoclinic class of #140094
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1919
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11293
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

449 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:

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

Polycyclic presentation:

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

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, 56285);