#7604474

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563327
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#532	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=h,d^2=g,e^2=fi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7511670);