#496146

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂³	C₂×#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

480 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 266 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=eg,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403342);