#412149

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1596 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:

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

Polycyclic presentation:

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

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, 319345);