#149011

order	512
nilpotency class	4
isoclinism	isoclinic class of #13490
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2073
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,d^b=di,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, 56207);