#412616

order	512
nilpotency class	3
isoclinism	isoclinic class of #400512
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2491
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15132
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

857 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,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, 319812);