#439019

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
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×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1451 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:

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

Polycyclic presentation:

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

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, 346215);