#462446

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

879 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369642);