#451627

order	512
nilpotency class	3
isoclinism	isoclinic class of #451626
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16005
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

734 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:

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

Polycyclic presentation:

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

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, 358823);