#438622

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

779 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=eg,c^2=ef,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 345818);