#398442

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	#23393	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1658 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)(12,24)(13,27)(14,31)(15,21)(16,20)(17,23)(19,28)(22,36)(25,32)(26,33)(29,40)(30,35)(34,37)(38,39);(9,11,14,23,15,12,16,26)(10,17,20,18,21,33,31,24)(13,22,28,37,29,25,30,39)(19,32,27,34,35,36,40,38);(1,2,3,4)(9,12,15,11)(10,18,21,24)(13,25,29,22)(14,26,16,23)(17,20,33,31)(19,34,35,38)(27,36,40,32)(28,39,30,37);(6,8)(9,13,14,28,15,29,16,30)(10,19,20,27,21,35,31,40)(11,22,23,37,12,25,26,39)(17,32,18,34,33,36,24,38)

Polycyclic presentation:

2^9|b^2=e,c^2=f,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,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, 305638);