#439077

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1445 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:

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

Polycyclic presentation:

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

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, 346273);