#398488

order	512
nilpotency class	3
isoclinism	isoclinic class of #394378
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1050 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:

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

Polycyclic presentation:

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

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, 305684);