#506305

order	512
nilpotency class	4
isoclinism	isoclinic class of #506304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#28165
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13334	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

548 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)(4,18)(5,11)(6,9)(7,27)(10,20)(12,26)(13,24)(14,29)(15,28)(16,45)(17,31)(19,46)(21,47)(22,39)(23,35)(25,33)(30,43)(32,44)(34,52)(36,56)(37,50)(38,63)(40,57)(41,59)(42,61)(48,55)(49,53)(51,60)(54,64)(58,62);(1,3)(2,8,11,31)(4,19,20,50)(5,17)(6,25)(7,16)(9,33,35,28)(10,37)(12,40,36,30)(13,32,27,45)(14,22,42,53)(15,23)(18,46)(21,54,52,51)(24,44)(26,57)(29,49)(34,60)(38,62)(39,61)(41,63,58,55)(43,56)(47,64)(48,59);(2,9)(6,26)(7,24)(8,28)(10,34)(11,35)(12,13)(15,43)(16,44)(18,47)(19,50)(21,55)(22,53)(23,56)(25,57)(27,36)(29,59)(30,32)(31,33)(37,60)(38,39)(40,45)(41,51)(46,64)(48,49)(52,63)(54,58)(61,62);(1,4)(2,10)(3,14,17,42)(5,20)(6,21)(7,22)(8,29,31,61)(9,34)(11,18)(12,38)(13,39)(15,41,25,58)(16,19,44,50)(23,52)(24,53)(26,55)(27,49)(28,59,33,62)(30,60,40,64)(32,37,45,46)(35,47)(36,48)(43,51,57,54)(56,63)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413501);