#418972

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3013	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2242 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:

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

Polycyclic presentation:

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

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, 326168);