#149017

order	512
nilpotency class	4
isoclinism	isoclinic class of #135851
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1854
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=h,e^2=hi,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,f^b=fi,g^b=gi,e^c=ei,f^c=fi

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, 56213);