#438295

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1178 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:

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

Polycyclic presentation:

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

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, 345491);