#398454

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
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	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1602 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)(12,29)(13,33)(14,21)(15,20)(16,22)(17,26)(18,25)(19,34)(24,35)(27,37)(28,36)(30,38)(31,40)(32,39);(9,11,14,26,16,28,15,27)(10,17,20,23,22,37,21,36)(12,30,31,34,32,13,25,24)(18,38,39,35,40,33,29,19);(1,2,3,4)(9,12,15,25,16,32,14,31)(10,18,21,29,22,40,20,39)(11,24,27,13,28,34,26,30)(17,35,36,38,37,19,23,33);(6,8)(9,13,14,24,16,30,15,34)(10,19,20,38,22,35,21,33)(11,25,26,12,28,31,27,32)(17,18,23,39,37,40,36,29)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305650);