#513178

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

431 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)(5,6,13,8)(7,19,23,54)(9,16,31,26)(10,15,35,25)(11,36,39,61)(12,28,38,18)(14,42,24,20)(17,49,27,33)(21,56,57,68)(22,58,40,53)(29,46,34,52)(30,45,41,48)(32,50,65,62)(37,47,67,60)(43,66,59,63)(44,55,51,64);(5,7,13,23)(6,14,8,24)(9,20,31,42)(10,32,35,65)(11,37,39,67)(12,40,38,22)(15,19,25,54)(16,47,26,60)(17,50,27,62)(18,51,28,44)(21,49,57,33)(29,63,34,66)(30,64,41,55)(36,43,61,59)(45,56,48,68)(46,58,52,53);(2,4)(5,8,13,6)(7,14,23,24)(9,25,31,15)(10,26,35,16)(11,27,39,17)(12,28,38,18)(19,42,54,20)(21,43,57,59)(22,44,40,51)(29,48,34,45)(30,52,41,46)(32,47,65,60)(33,61,49,36)(37,50,67,62)(53,64,58,55)(56,66,68,63);(5,9)(6,15)(8,25)(10,11)(12,41)(13,31)(16,17)(18,52)(19,47)(20,32)(21,37)(22,40)(26,27)(28,46)(29,33)(30,38)(34,49)(35,39)(36,45)(42,65)(43,50)(44,51)(48,61)(53,68)(54,60)(55,66)(56,58)(57,67)(59,62)(63,64)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^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, 420374);