#513176

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
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

447 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^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, 420372);