#397498

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1313 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:

(1,2,4,3)(5,6)(7,8)(9,10,15,21)(11,18,12,24)(13,27,29,40)(14,31,16,20)(17,26,33,23)(19,30,35,28)(22,36,25,32)(34,39,38,37);(1,3,4,2)(9,11,16,26,15,12,14,23)(10,17,31,24,21,33,20,18)(13,22,30,39,29,25,28,37)(19,32,40,38,35,36,27,34);(2,3)(9,12)(10,18)(11,15)(13,25)(14,26)(16,23)(17,31)(19,34)(20,33)(21,24)(22,29)(27,36)(28,39)(30,37)(32,40)(35,38);(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|a^2=f,b^2=ef,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, 304694);