#451214

order	512
nilpotency class	3
isoclinism	isoclinic class of #427241
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15989
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

749 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)(3,9,16,34)(4,19,24,47)(5,25,28,11)(6,13)(7,30)(8,12)(10,22,37,23)(14,35)(15,32)(17,33,18,40)(20,31)(21,46)(26,53)(27,39,29,38)(36,50)(41,63,55,60)(42,61)(43,57)(44,58)(45,59)(48,64,49,56)(51,62)(52,54);(1,3,6,18,8,16,7,17)(2,9,30,33,12,34,13,40)(4,20,22,42,24,43,23,14)(5,26,27,44,28,15,29,45)(10,35,47,57,37,61,19,31)(11,32,39,58,25,53,38,59)(21,49,50,55,51,48,52,41)(36,60,46,56,54,63,62,64);(1,4)(2,10,12,37)(3,14,16,42)(5,21)(6,22)(7,23)(8,24)(9,31)(11,36,25,54)(13,19,30,47)(15,41,26,55)(17,43,18,20)(27,50)(28,51)(29,52)(32,56)(33,35)(34,57)(38,62,39,46)(40,61)(44,48,45,49)(53,64)(58,63)(59,60);(1,5)(2,11)(3,15)(4,21)(6,29)(7,27)(8,28)(9,32)(10,36)(12,25)(13,39)(14,41)(16,26)(17,45)(18,44)(19,46)(20,48)(22,52)(23,50)(24,51)(30,38)(31,56)(33,59)(34,53)(35,60)(37,54)(40,58)(42,55)(43,49)(47,62)(57,64)(61,63)

Polycyclic presentation:

2^9|a^2=fgh,b^2=ef,c^2=e,f^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,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358410);