#149015

order	512
nilpotency class	4
isoclinism	isoclinic class of #135678
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1835
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11268
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

423 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=h,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=di,g^b=gi,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, 56211);