#219502

order	512
nilpotency class	3
isoclinism	isoclinic class of #219498
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#518
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

567 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=efh,d^2=eh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cei,f^b=fh,g^c=gh

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, 126698);