#219534

order	512
nilpotency class	3
isoclinism	isoclinic class of #219532
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	C₂×#496	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

607 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,7,4,6,3,8)(9,10)(11,13)(12,25,30,39)(14,33)(15,20)(16,21)(17,23)(18,29,36,28)(19,31)(22,34,37,38)(24,32)(26,40,35,27);(1,2,3,8,4,6,5,7)(9,11,14,23,16,24,15,19)(10,17,20,13,21,31,33,32)(12,26,29,22,30,35,28,37)(18,27,39,34,36,40,25,38);(2,6)(7,8)(9,12,16,30)(10,18)(11,22)(13,27,32,40)(14,28,15,29)(17,34,31,38)(19,35)(20,25)(21,36)(23,26)(24,37)(33,39);(1,3,4,5)(2,7,6,8)(9,13,14,31,16,32,15,17)(10,19,20,11,21,23,33,24)(12,27,28,38,30,40,29,34)(18,35,25,22,36,26,39,37)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=efh,d^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,d^b=di,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, 126730);