#459368

order	512
nilpotency class	3
isoclinism	isoclinic class of #454109
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2670
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16339
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

784 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366564);