#219558

order	512
nilpotency class	3
isoclinism	isoclinic class of #219552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	#3147	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

653 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)(3,6)(7,8)(9,10)(11,22)(12,28,32,42)(13,33)(14,20)(15,21)(16,25)(17,31,43,57)(18,35)(19,34)(23,48,51,37)(24,52)(26,39)(27,46)(29,40,50,65)(30,41,60,66)(36,44)(38,55)(45,62)(47,53)(49,69,71,64)(54,67)(56,72,61,59)(58,68,70,63);(9,11,14,25,15,27,34,26)(10,16,19,22,21,39,20,46)(12,29,57,23,32,50,31,51)(13,24,35,53,36,55,62,54)(17,40,28,37,43,65,42,48)(18,38,33,47,45,52,44,67)(30,58,72,49,60,70,59,71)(41,64,56,63,66,69,61,68);(1,2,6,8,4,7,3,5)(9,12,15,32)(10,17)(11,23)(13,30,36,60)(14,31,34,57)(16,37,39,48)(18,41)(19,42)(20,28)(21,43)(22,40,46,65)(24,49)(25,29)(26,50)(27,51)(33,61)(35,59,62,72)(38,63,52,68)(44,56)(45,66)(47,64,67,69)(53,58)(54,70)(55,71);(9,13,34,62,15,36,14,35)(10,18,20,44,21,45,19,33)(11,24,26,54,27,55,25,53)(12,30,57,72,32,60,31,59)(16,38,46,67,39,52,22,47)(17,41,28,56,43,66,42,61)(23,49,50,70,51,71,29,58)(37,63,65,69,48,68,40,64)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=efi,d^2=eh,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,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, 126754);