#149090

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	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

441 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,f^2=h,g^2=hi,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, 56286);