#419888

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1570 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)(5,6)(7,8)(9,10)(11,23)(12,18)(13,30)(14,21)(15,20)(16,22)(17,26)(19,33)(24,25)(27,37)(28,36)(29,32)(31,34)(35,38)(39,40);(9,11,16,28)(10,17,22,37)(12,29,19,38)(13,32,34,35)(14,27,15,26)(18,25,33,40)(20,36,21,23)(24,30,39,31);(2,4)(9,12,16,19)(10,18,22,33)(11,24,28,39)(13,20,34,21)(14,30,15,31)(17,35,37,32)(23,25,36,40)(26,38,27,29);(6,8)(9,13,15,33,16,34,14,18)(10,19,21,30,22,12,20,31)(11,25,27,35,28,40,26,32)(17,24,36,29,37,39,23,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=ei,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,g^b=gi,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, 327084);