#496028

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

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	#532	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ 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)(9,10,15,22,16,21,35,52,17,23,34,19,14,42,12,31)(11,24,28,53,29,45,25,46,30,55,57,62,27,18,32,43)(13,20,39,49,40,50,37,67,41,51,64,47,38,68,33,63)(26,54,60,71,61,48,56,70,36,65,66,72,59,44,58,69);(2,5)(6,8)(9,11,14,27,17,30,16,29)(10,18,21,24,23,45,42,55)(12,32,34,57,35,25,15,28)(13,36,38,61,41,26,40,59)(19,46,31,62,22,43,52,53)(20,48,50,65,51,44,68,54)(33,56,64,60,37,58,39,66)(47,69,63,71,49,70,67,72);(1,3,4,7)(2,6,5,8)(9,12,14,34,17,35,16,15)(10,19,21,31,23,22,42,52)(11,25,27,28,30,32,29,57)(13,37,38,39,41,33,40,64)(18,43,24,53,45,46,55,62)(20,49,50,67,51,47,68,63)(26,58,59,66,36,56,61,60)(44,70,54,72,48,69,65,71);(9,13,15,39,16,40,35,37,17,41,34,64,14,38,12,33)(10,20,22,49,21,50,52,67,23,51,19,47,42,68,31,63)(11,26,28,60,29,61,25,56,30,36,57,66,27,59,32,58)(18,44,43,69,24,54,53,71,45,48,46,70,55,65,62,72)

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,d^b=di,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, 403224);