#6685516

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
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	#60773	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1747 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,27)(13,19)(14,20)(15,21)(16,22)(18,31)(23,43)(24,39)(25,40)(26,42)(28,56)(29,46)(30,57)(32,45)(33,47)(34,48)(35,50)(36,41)(37,49)(38,52)(44,60)(51,66)(53,63)(54,64)(55,65)(58,71)(59,67)(61,68)(62,69)(70,72);(2,5)(3,7)(9,11)(10,17)(12,28)(13,24)(14,34)(15,36)(16,26)(18,43)(19,39)(20,48)(21,41)(22,42)(23,31)(25,33)(27,56)(29,53)(30,51)(32,52)(35,54)(37,55)(38,45)(40,47)(44,62)(46,63)(49,65)(50,64)(57,66)(58,59)(60,69)(61,70)(67,71)(68,72);(9,12,13,31,16,32,15,29)(10,18,21,27,22,46,19,45)(11,23,24,52,26,53,36,28)(14,30,33,60,35,61,37,59)(17,38,41,43,42,56,39,63)(20,44,49,57,50,67,47,68)(25,51,54,69,55,70,34,58)(40,62,48,66,65,71,64,72);(2,5)(6,8)(9,13,16,15)(10,19,22,21)(11,24,26,36)(12,29,32,31)(14,33,35,37)(17,39,42,41)(18,27,46,45)(20,47,50,49)(23,28,53,52)(25,54,55,34)(30,59,61,60)(38,43,56,63)(40,64,65,48)(44,57,67,68)(51,58,70,69)(62,66,71,72);(1,3,4,7)(2,6,5,8)(9,14)(10,20)(11,25)(12,30)(13,33)(15,37)(16,35)(17,40)(18,44)(19,47)(21,49)(22,50)(23,51)(24,54)(26,55)(27,57)(28,58)(29,59)(31,60)(32,61)(34,36)(38,62)(39,64)(41,48)(42,65)(43,66)(45,68)(46,67)(52,69)(53,70)(56,71)(63,72)

Polycyclic presentation:

2^9|c^2=g,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592712);