#149106

order	512
nilpotency class	3
isoclinism	isoclinic class of #2104
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#401
commutator, prime radical	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11298
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

423 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56302);