#446015

order	512
nilpotency class	3
isoclinism	isoclinic class of #419458
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2709
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15828
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

763 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:

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

Polycyclic presentation:

2^9|a^2=efg,c^2=efg,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,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, 353211);