#148896

order	512
nilpotency class	4
isoclinism	isoclinic class of #134702
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1809
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11232
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

457 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,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=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,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, 56092);