#433971

order	512
nilpotency class	3
isoclinism	isoclinic class of #433970
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

966 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:

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

Polycyclic presentation:

2^9|a^2=eg,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,g^b=gi,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, 341167);