#7597046

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563333
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	#61081	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)(13,21,41,54)(14,20,40,55)(15,23,16,22)(18,30,47,29)(19,36,52,37)(26,58,61,43)(27,45,28,44)(33,48,60,69)(34,65,66,49)(35,50,42,56)(39,53,62,57)(59,72,63,71)(64,70,67,68);(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)(13,39,40,27,41,62,14,28)(19,48,51,58,52,69,32,43)(20,53,54,44,55,57,21,45)(34,59,35,64,66,63,42,67)(49,68,50,71,65,70,56,72);(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)(13,34,40,35,41,66,14,42)(18,43,46,69,47,58,25,48)(20,49,54,50,55,65,21,56)(27,59,39,67,28,63,62,64)(44,68,53,72,45,70,57,71);(1,3,4,8)(2,6,5,7)(9,13,15,40,17,41,16,14)(10,20,22,54,24,55,23,21)(11,27,30,39,31,28,29,62)(12,34,36,35,38,66,37,42)(18,44,46,53,47,45,25,57)(19,49,51,50,52,65,32,56)(26,59,60,67,61,63,33,64)(43,68,69,72,58,70,48,71);(9,14,16,41,17,40,15,13)(10,21,23,55,24,54,22,20)(11,28,30,62,31,27,29,39)(12,35,37,34,38,42,36,66)(18,45,46,57,47,44,25,53)(19,50,32,49,52,56,51,65)(26,59,60,67,61,63,33,64)(43,68,69,72,58,70,48,71)

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,e^c=ei

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, 7504242);