#513167

order	512
nilpotency class	5
isoclinism	isoclinic class of D₄×D₃₂
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

444 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)(7,17,22,50)(8,23,27,48)(9,28,31,47)(10,32,30,15)(12,34,37,19)(13,38,40,25)(14,41,45,24)(18,39,53,66)(20,55,56,68)(21,57,33,49)(26,62,60,59)(29,42,64,67)(35,58,65,52)(36,54,46,51)(43,63,44,61);(5,7,11,22)(6,12,16,37)(8,18,27,53)(9,29,31,64)(10,33,30,21)(13,39,40,66)(14,42,45,67)(15,46,32,36)(17,48,50,23)(19,41,34,24)(20,38,56,25)(26,58,60,52)(28,35,47,65)(43,55,44,68)(49,59,57,62)(51,63,54,61);(2,4);(6,13)(7,18)(8,24)(9,25)(10,30)(12,17)(14,43)(15,47)(16,40)(19,20)(21,58)(22,53)(26,63)(27,41)(28,32)(29,51)(31,38)(33,52)(34,56)(35,55)(36,57)(37,50)(39,42)(44,45)(46,49)(54,64)(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,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

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420363);