#419859

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1616 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)(9,10)(11,20)(12,17)(13,25)(14,29)(15,19)(16,21)(18,28)(22,35)(23,31)(24,37)(26,32)(27,34)(30,33)(36,38)(39,40);(5,6)(7,8)(9,11)(10,16)(12,21)(13,26)(14,30)(15,23)(17,31)(18,35)(19,33)(20,29)(22,27)(24,36)(25,38)(28,39)(32,34)(37,40);(2,4)(6,8)(9,12,15,14)(10,17,19,29)(11,21,23,30)(13,24,28,27)(16,31,33,20)(18,34,25,37)(22,26,36,39)(32,38,40,35);(9,13,14,27,15,28,12,24)(10,18,17,34,19,25,29,37)(11,22,21,26,23,36,30,39)(16,32,20,35,33,40,31,38)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,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, 327055);