#512945

order	512
nilpotency class	5
isoclinism	isoclinic class of #512941
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 ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

432 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 68
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232017, #72321829, #72321831, …
maximal normal subgroup of	#72232017, #72321829, #72321831, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420141);