#149217

order	512
nilpotency class	4
isoclinism	isoclinic class of #13334
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2011
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11324
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

509 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=e,d^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=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, 56413);