#504288

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
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

485 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)(11,26,31,49)(12,32,30,44)(13,36,40,67)(14,22,33,46)(15,21,42,47)(16,23,41,24)(18,35,19,28)(20,51,55,37)(27,57,61,71)(29,50,62,45)(34,48,60,43)(38,53,69,54)(39,66,68,52)(56,64,65,58)(59,63,72,70);(2,5)(6,7)(9,11,15,29,16,12,33,35,17,31,42,62,41,30,14,28)(10,18,22,44,24,45,47,49,25,19,46,32,23,50,21,26)(13,27,38,59,39,34,51,64,40,61,69,72,68,60,37,58)(20,43,52,70,54,71,67,65,55,48,66,63,53,57,36,56);(9,12,33,62,41,11,15,35,17,30,14,29,16,31,42,28)(10,19,47,26,23,45,22,32,25,18,21,49,24,50,46,44)(13,34,51,72,68,27,38,64,40,60,37,59,39,61,69,58)(20,48,67,56,53,71,52,63,55,43,36,65,54,57,66,70);(1,3,4,8)(2,6,5,7)(9,13,15,38,16,39,33,51,17,40,42,69,41,68,14,37)(10,20,22,52,24,54,47,67,25,55,46,66,23,53,21,36)(11,27,29,59,12,34,35,64,31,61,62,72,30,60,28,58)(18,43,44,70,45,71,49,65,19,48,32,63,50,57,26,56)

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,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411484);