#496027

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

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	#532	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

526 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=g,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,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, 403223);