#499381

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#466
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

678 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)(3,6)(4,5)(7,8)(9,10,14,19,33,46,63,70,15,20,35,48,13,18,32,45)(11,16,23,38,24,39,55,68,25,40,56,69,22,37,53,66)(12,26,29,42,28,58,61,44,31,59,62,43,30,47,34,17)(21,49,50,41,27,57,60,65,52,71,72,64,51,67,54,36);(2,5)(6,8)(9,11)(10,16)(12,27)(13,22)(14,23)(15,25)(17,41)(18,37)(19,38)(20,40)(21,30)(24,33)(26,57)(28,52)(29,60)(31,51)(32,53)(34,50)(35,56)(36,43)(39,46)(42,65)(44,64)(45,66)(47,49)(48,69)(54,62)(55,63)(58,71)(59,67)(61,72)(68,70);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,28)(14,34)(15,31)(16,36)(18,42)(19,47)(20,44)(22,27)(23,54)(24,51)(25,52)(26,45)(29,32)(30,33)(35,61)(37,41)(38,67)(39,64)(40,65)(43,46)(48,58)(49,66)(50,53)(55,72)(56,60)(57,69)(59,70)(62,63)(68,71);(9,13,15,33)(10,18,20,46)(11,22,25,24)(12,28,31,30)(14,32,35,63)(16,37,40,39)(17,42,44,43)(19,45,48,70)(21,27,52,51)(23,53,56,55)(26,58,59,47)(29,61,62,34)(36,41,65,64)(38,66,69,68)(49,57,71,67)(50,60,72,54)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406577);