#411947

order	512
nilpotency class	3
isoclinism	isoclinic class of #395246
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1112 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,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, 319143);