#423662

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

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

1557 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)(11,23)(12,13)(14,22)(15,20)(16,21)(17,26)(18,19)(24,25)(27,36)(28,37)(29,35)(30,34)(31,33)(32,40)(38,39);(9,11,16,28)(10,17,21,36)(12,29,30,24)(13,32,34,39)(14,27,15,26)(18,35,33,25)(19,38,31,40)(20,37,22,23);(1,2,3,4)(9,12,14,19,16,30,15,31)(10,18,20,13,21,33,22,34)(11,24,26,38,28,29,27,40)(17,35,23,39,36,25,37,32);(6,8)(9,13,15,18,16,34,14,33)(10,19,22,12,21,31,20,30)(11,25,27,39,28,35,26,32)(17,24,37,40,36,29,23,38)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=ei,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=de,e^b=ei,f^b=fi,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, 330858);