#433976

order	512
nilpotency class	3
isoclinism	isoclinic class of #433975
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15494
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

973 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)(4,21,26,42)(5,27,31,11)(6,13)(7,12)(8,14)(9,18)(10,25,38,24)(16,44,47,32)(17,39,49,43)(19,35)(20,34)(22,36,46,59)(23,50)(28,33,48,58)(29,40,30,41)(37,53)(45,63)(51,60)(52,55)(54,61)(56,62)(57,64);(1,3,8,20)(2,9,14,35)(4,22,26,46)(5,28,31,48)(6,19,7,18)(10,36,38,59)(11,39,27,43)(12,34,13,15)(16,25,47,24)(17,30,49,29)(21,32,42,44)(23,45,54,64)(33,41,58,40)(37,57,55,63)(50,62,61,60)(51,53,56,52);(1,4)(2,10,14,38)(3,16)(5,23)(6,24)(7,25)(8,26)(9,32,35,44)(11,37,27,55)(12,21,13,42)(15,36,34,59)(17,45)(18,22)(19,46)(20,47)(28,56)(29,52)(30,53)(31,54)(33,57,58,63)(39,62,43,60)(40,61,41,50)(48,51)(49,64);(1,5)(2,11)(3,17)(4,23)(6,30)(7,29)(8,31)(9,33)(10,37)(12,41)(13,40)(14,27)(15,43)(16,45)(18,48)(19,28)(20,49)(21,50)(22,51)(24,53)(25,52)(26,54)(32,57)(34,39)(35,58)(36,60)(38,55)(42,61)(44,63)(46,56)(47,64)(59,62)

Polycyclic presentation:

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

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, 341172);