#459924

order	512
nilpotency class	3
isoclinism	isoclinic class of #459923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16352
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

891 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367120);