#432669

order	512
nilpotency class	3
isoclinism	isoclinic class of #432668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15463
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

1073 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:

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

Polycyclic presentation:

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

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, 339865);