#7597040

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	#61086	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1940 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)(3,7)(4,5)(6,8)(9,10,17,24)(11,25,31,46)(12,32,38,51)(13,21,42,55)(14,20,41,56)(15,23,16,22)(18,30,47,29)(19,36,52,37)(26,58,61,43)(27,45,28,44)(33,48,60,68)(34,54,40,49)(35,50,66,64)(39,53,63,57)(59,71,62,70)(65,69,72,67);(1,2)(3,6)(4,5)(7,8)(9,11,15,29,17,31,16,30)(10,18,22,25,24,47,23,46)(12,33,36,61,38,60,37,26)(13,39,41,27,42,63,14,28)(19,48,51,58,52,68,32,43)(20,53,55,44,56,57,21,45)(34,59,35,65,40,62,66,72)(49,67,50,70,54,69,64,71);(2,5)(6,7)(9,12,15,36,17,38,16,37)(10,19,22,51,24,52,23,32)(11,26,30,60,31,61,29,33)(13,40,41,66,42,34,14,35)(18,43,46,68,47,58,25,48)(20,54,55,64,56,49,21,50)(27,62,39,65,28,59,63,72)(44,69,53,70,45,67,57,71);(1,3,4,8)(2,6,5,7)(9,13,15,41,17,42,16,14)(10,20,22,55,24,56,23,21)(11,27,30,39,31,28,29,63)(12,34,36,35,38,40,37,66)(18,44,46,53,47,45,25,57)(19,49,51,50,52,54,32,64)(26,59,60,72,61,62,33,65)(43,67,68,71,58,69,48,70);(9,14,16,42,17,41,15,13)(10,21,23,56,24,55,22,20)(11,28,30,63,31,27,29,39)(12,35,37,34,38,66,36,40)(18,45,46,57,47,44,25,53)(19,50,32,49,52,64,51,54)(26,59,60,72,61,62,33,65)(43,67,68,71,58,69,48,70)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=gi,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, 7504236);