#489391

order	512
nilpotency class	4
isoclinism	isoclinic class of #489390
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#62433	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

752 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=g,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dgi,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396587);