#412150

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

Polycyclic presentation:

2^9|b^2=e,c^2=e,d^2=hi,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, 319346);