#219576

order	512
nilpotency class	3
isoclinism	isoclinic class of #219576
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₄² ▷ C₂² ▷ 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=e,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=dh,f^b=fh,g^c=gh

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, 126772);