#451760

order	512
nilpotency class	3
isoclinism	isoclinic class of #451759
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16012
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

871 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)(4,19,22,36)(5,23,26,39)(6,12)(7,13)(8,17)(9,21,37,55)(10,27,40,25)(11,29)(15,30,44,34)(16,45,46,31)(18,32)(20,35)(24,38,47,58)(28,33)(41,43,48,54)(42,62)(49,51)(50,57)(52,56)(53,61)(59,63)(60,64);(1,3,7,18)(2,8,13,33)(4,15,22,44)(5,24,26,47)(6,28,29,17)(9,34,37,30)(10,38,40,58)(11,32,12,14)(16,27,46,25)(19,48,36,41)(20,50,53,63)(21,54,55,43)(23,31,39,45)(35,60,61,62)(42,52,64,51)(49,57,56,59);(1,4,6,21,7,22,29,55)(2,9,11,36,13,37,12,19)(3,15,17,43,18,44,28,54)(5,20,25,51,26,53,27,52)(8,30,14,41,33,34,32,48)(10,35,23,56,40,61,39,49)(16,42,24,50,46,64,47,63)(31,57,38,62,45,59,58,60);(1,5,6,25,7,26,29,27)(2,10,12,39,13,40,11,23)(3,16,17,24,18,46,28,47)(4,20,21,51,22,53,55,52)(8,31,32,58,33,45,14,38)(9,35,19,49,37,61,36,56)(15,42,43,50,44,64,54,63)(30,57,48,60,34,59,41,62)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358956);