#401566

order	512
nilpotency class	3
isoclinism	isoclinic class of #394483
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2472
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14819
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

825 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:

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

Polycyclic presentation:

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

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, 308762);