#513180

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#33851	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,12,18)(7,19,20,25)(8,9,28,33)(10,34,38,66)(11,39,37,17)(13,42,21,22)(14,15,47,36)(16,49,30,31)(23,58,54,62)(24,60,40,53)(26,46,63,52)(27,45,41,32)(29,50,61,59)(35,48,68,56)(43,67,55,65)(44,57,51,64);(5,7,12,20)(6,13,18,21)(8,25,28,19)(9,29,33,61)(10,35,38,68)(11,40,37,24)(14,22,47,42)(15,48,36,56)(16,50,30,59)(17,51,39,44)(23,49,54,31)(26,62,63,58)(27,53,41,60)(32,67,45,65)(34,43,66,55)(46,64,52,57);(2,4)(7,20)(13,21)(19,25)(22,42)(23,54)(24,40)(29,61)(35,68)(43,55)(44,51)(48,56)(50,59)(53,60)(57,64)(58,62)(65,67);(5,8,12,28)(6,14,18,47)(7,21,20,13)(9,30,33,16)(10,36,38,15)(11,41,37,27)(17,52,39,46)(19,29,25,61)(22,56,42,48)(23,59,54,50)(24,44,40,51)(26,49,63,31)(32,34,45,66)(35,55,68,43)(53,67,60,65)(57,58,64,62)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,g^a=gh,h^a=hi,c^b=ci,d^b=de,e^b=eg,g^b=gh,h^b=hi,e^d=eg,g^d=gh,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420376);