#417982

order	512
nilpotency class	3
isoclinism	isoclinic class of #417981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15259
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1919 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)(3,8)(4,17,20,9)(5,22)(6,25)(7,11)(10,24)(12,27)(13,36,18,28)(14,39)(15,30)(16,43)(19,49,50,54)(21,33,52,46)(23,35)(26,34)(29,40)(31,42)(32,51,45,53)(37,61,47,64)(38,56,48,60)(41,58)(44,57)(55,62,59,63);(1,3)(2,8)(4,18,20,13)(5,14)(6,16)(7,15)(9,28,17,36)(10,29)(11,30)(12,31)(19,37,50,47)(21,48,52,38)(22,39)(23,41)(24,40)(25,43)(26,44)(27,42)(32,59,45,55)(33,56,46,60)(34,57)(35,58)(49,64,54,61)(51,62,53,63);(1,4)(2,9)(3,13)(5,19)(6,21)(7,20)(8,28)(10,32)(11,17)(12,33)(14,37)(15,18)(16,38)(22,54)(23,50)(24,51)(25,46)(26,53)(27,52)(29,55)(30,36)(31,56)(34,45)(35,49)(39,64)(40,62)(41,47)(42,48)(43,60)(44,63)(57,59)(58,61);(1,5,7,23)(2,10,11,34)(3,14,15,41)(4,19,20,50)(6,26,27,24)(8,29,30,57)(9,32,17,45)(12,35,25,22)(13,37,18,47)(16,44,42,40)(21,53,52,51)(28,55,36,59)(31,58,43,39)(33,49,46,54)(38,63,48,62)(56,61,60,64)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286066, #72286176, #72286267
maximal normal subgroup of	#72286066, #72286176, #72286267

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325178);