#412096

order	512
nilpotency class	3
isoclinism	isoclinic class of #394335
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319292);