#513031

order	512
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂²	C₂²×D₁₆
prime radical	#33713	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

398 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,20)(12,19)(13,21)(14,23)(15,22)(16,43)(17,45)(18,25)(24,38)(26,50)(27,51)(28,63)(29,65)(30,52)(31,46)(32,47)(33,66)(34,49)(35,54)(36,53)(37,58)(39,67)(40,57)(41,56)(42,55)(44,68)(48,61)(59,70)(60,69)(62,71)(64,72);(2,5)(6,8)(9,11,15,27,16,28,35,31,13,26,36,59,37,60,40,34,18,30,42,62,44,64,39,33,17,29,41,61,38,32,14,12)(10,19,23,47,24,48,56,65,45,66,67,72,68,71,55,52,25,49,57,69,58,70,53,50,21,46,54,63,43,51,22,20);(1,3,4,7)(2,6,5,8)(9,12,14,32,38,61,41,29,17,33,39,64,44,62,42,30,18,34,40,60,37,59,36,26,13,31,35,28,16,27,15,11)(10,20,22,51,43,63,54,46,21,50,53,70,58,69,57,49,25,52,55,71,68,72,67,66,45,65,56,48,24,47,23,19);(9,13,18,17)(10,21,25,45)(11,26,30,29)(12,31,34,33)(14,35,40,39)(15,36,42,41)(16,37,44,38)(19,46,49,66)(20,50,52,65)(22,53,55,56)(23,54,57,67)(24,43,58,68)(27,59,62,61)(28,60,64,32)(47,63,69,72)(48,51,70,71)

Polycyclic presentation:

2^9|b^2=f,c^2=e,d^2=i,e^2=gh,f^2=g,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fgh,g^a=gh,h^a=hi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420227);