#148998

order	512
nilpotency class	4
isoclinism	isoclinic class of #13417
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2041
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11267
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

473 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)(3,14,18,37)(4,20)(5,25)(6,28)(7,13)(8,16,31,17)(9,22)(10,26)(11,38)(12,29)(15,42,21,54)(19,50,47,41)(23,34)(24,58)(27,39)(30,55,33,46)(32,49,59,48)(35,43)(36,60)(40,44,51,45)(52,56)(53,57)(61,63,64,62);(1,3,5,17,6,19,26,49,7,18,27,16,29,47,38,48)(2,8,11,14,12,32,39,41,13,31,10,37,28,59,25,50)(4,21,22,44,24,55,56,62,23,15,43,45,57,46,60,63)(9,33,20,51,36,54,53,64,35,30,34,40,52,42,58,61);(1,4,5,22,6,24,26,56,7,23,27,43,29,57,38,60)(2,9,10,34,12,36,25,58,13,35,11,20,28,52,39,53)(3,15,16,44,19,46,48,62,18,21,17,45,47,55,49,63)(8,30,14,40,32,42,41,61,31,33,37,51,59,54,50,64)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,d^b=di,g^b=gi,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, 56194);