#513125

order	512
nilpotency class	5
isoclinism	isoclinic class of #513098
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#13663	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|b^2=ei,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,h^a=hi,c^b=ce,d^b=di,f^c=fg,g^c=gh,h^c=hi

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, 420321);