#149007

order	512
nilpotency class	4
isoclinism	isoclinic class of #13545
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2101
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11268
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

427 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=h,e^2=h,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,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, 56203);