#6470822

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470818
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1916 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)(3,5)(4,6)(7,8)(9,10,16,21)(11,22,27,47)(12,28,32,18)(13,19,34,44)(14,20,15,36)(17,25,39,26)(23,48,52,45)(24,46,54,63)(29,57,60,40)(30,58,31,41)(33,42,35,43)(37,51,56,50)(38,55,64,53)(49,68,70,67)(59,66,61,65)(62,69,72,71);(1,3)(2,5)(4,7)(6,8)(9,11,14,25,16,27,15,26)(10,17,36,47,21,39,20,22)(12,23,30,50,32,52,31,51)(13,24,35,55,34,54,33,53)(18,37,58,48,28,56,41,45)(19,38,42,63,44,64,43,46)(29,49,61,71,60,70,59,69)(40,62,65,68,57,72,66,67);(3,7)(5,8)(9,12,15,31,16,32,14,30)(10,18,20,41,21,28,36,58)(11,23,26,51,27,52,25,50)(13,29,33,59,34,60,35,61)(17,37,22,45,39,56,47,48)(19,40,43,66,44,57,42,65)(24,49,53,69,54,70,55,71)(38,62,46,67,64,72,63,68);(3,7)(5,8)(9,13)(10,19)(11,24)(12,29)(14,35)(15,33)(16,34)(17,38)(18,40)(20,43)(21,44)(22,46)(23,49)(25,55)(26,53)(27,54)(28,57)(30,61)(31,59)(32,60)(36,42)(37,62)(39,64)(41,66)(45,67)(47,63)(48,68)(50,71)(51,69)(52,70)(56,72)(58,65);(2,6)(5,8)(9,14,16,15)(10,20,21,36)(11,25,27,26)(12,30,32,31)(13,33,34,35)(17,22,39,47)(18,41,28,58)(19,42,44,43)(23,50,52,51)(24,53,54,55)(29,59,60,61)(37,45,56,48)(38,63,64,46)(40,65,57,66)(49,69,70,71)(62,68,72,67)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,g^c=gi,e^d=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, 6378018);