#466735

order	512
nilpotency class	3
isoclinism	isoclinic class of #457096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1346 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)(3,4)(6,7)(9,10)(11,21)(12,26)(13,19)(14,17)(15,20)(16,24)(18,29)(22,32)(23,31)(25,33)(27,35)(28,36)(30,34)(37,38)(39,40);(9,11)(10,16,20,33)(12,27)(13,24)(14,25)(15,22)(17,21,19,32)(18,35,36,31)(23,30)(26,37,34,40)(28,38)(29,39);(5,6,8,7)(10,17)(11,22)(12,28)(13,14)(16,21)(18,36)(19,20)(23,38)(27,39)(29,30)(31,35)(32,33);(2,4)(5,7,8,6)(9,12,13,28,15,30,14,29)(10,18,17,34,20,36,19,26)(11,23,24,39,22,27,25,38)(16,31,21,37,33,35,32,40)

Polycyclic presentation:

2^9|b^2=gi,c^2=ef,d^2=fi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,f^c=fi,h^c=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, 373931);