#504291

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	#576
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

483 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=fh,d^2=eh,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411487);