#475398

order	512
nilpotency class	3
isoclinism	isoclinic class of #475397
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17046
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

659 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382594);