#438623

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

Polycyclic presentation:

2^9|b^2=egi,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, 345819);