#496144

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×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×C₁₆ ◁ 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)(9,10,15,22,16,21,34,49,17,23,33,19,14,39,12,30)(11,18,27,40,28,42,24,50,29,43,52,44,26,57,31,58)(13,20,36,47,37,46,62,68,38,48,61,45,35,63,32,59)(25,41,54,64,55,65,51,69,56,66,70,67,53,71,60,72);(1,3,5,4)(2,6,7,8)(9,11)(10,18)(12,31)(13,25)(14,26)(15,27)(16,28)(17,29)(19,44)(20,41)(21,42)(22,40)(23,43)(24,34)(30,58)(32,60)(33,52)(35,53)(36,54)(37,55)(38,56)(39,57)(45,67)(46,65)(47,64)(48,66)(49,50)(51,62)(59,72)(61,70)(63,71)(68,69);(9,12,14,33,17,34,16,15)(10,19,21,30,23,22,39,49)(11,24,26,27,29,31,28,52)(13,32,35,61,38,62,37,36)(18,40,42,50,43,44,57,58)(20,45,46,59,48,47,63,68)(25,51,53,54,56,60,55,70)(41,64,65,69,66,67,71,72);(1,4,5,3)(2,6,7,8)(9,13,15,36,16,37,34,62,17,38,33,61,14,35,12,32)(10,20,22,47,21,46,49,68,23,48,19,45,39,63,30,59)(11,25,27,54,28,55,24,51,29,56,52,70,26,53,31,60)(18,41,40,64,42,65,50,69,43,66,44,67,57,71,58,72)

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

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, 403340);