#419115

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1860 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=i,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^b=gi,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, 326311);