#425015

order	512
nilpotency class	3
isoclinism	isoclinic class of #417630
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15327
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1045 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)(3,15,19,34)(4,10)(5,26,29,40)(6,12)(7,13)(8,14)(9,18,33,20)(11,30,41,28)(16,44,47,35)(17,48)(21,51,46,31)(22,53,56,58)(23,37)(24,42)(25,38)(27,43)(32,50)(36,55,61,57)(39,49)(45,63)(52,60)(54,59)(62,64);(1,3)(2,9)(4,21)(5,27)(6,20)(7,18)(8,19)(10,35)(11,39)(12,34)(13,15)(14,33)(16,24)(17,28)(22,54)(23,47)(25,46)(26,48)(29,49)(30,50)(31,37)(32,40)(36,59)(38,44)(41,43)(42,51)(45,56)(52,55)(53,60)(57,64)(58,62)(61,63);(1,4,6,23,8,25,7,24)(2,10,12,37,14,38,13,42)(3,16,20,21,19,47,18,46)(5,22,30,57,29,56,28,55)(9,31,34,44,33,51,15,35)(11,36,26,58,41,61,40,53)(17,45,27,52,50,54,49,64)(32,59,39,62,48,63,43,60);(1,5)(2,11)(3,17)(4,22)(6,30)(7,28)(8,29)(9,32)(10,36)(12,26)(13,40)(14,41)(15,43)(16,45)(18,49)(19,50)(20,27)(21,52)(23,57)(24,55)(25,56)(31,59)(33,48)(34,39)(35,60)(37,58)(38,61)(42,53)(44,62)(46,64)(47,54)(51,63)

Polycyclic presentation:

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

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, 332211);