#512990

order	512
nilpotency class	5
isoclinism	isoclinic class of #13717
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2233
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂	C₂×#2233
prime radical	#33604	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

363 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|c^2=eh,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,f^b=fi,e^c=ei,f^c=fgi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420186);