#419022

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
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

2238 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)(9,10,14,20,15,21,33,19)(11,22,24,16,25,34,40,35)(12,26,28,37,29,39,27,17)(13,30,32,38,23,36,31,18);(1,2,3,4)(5,6)(7,8)(9,11,14,24,15,25,33,40)(10,16,19,22,21,35,20,34)(12,13,27,31,29,23,28,32)(17,36,26,18,37,30,39,38);(6,8)(9,12,15,29)(10,17,21,37)(11,23,25,13)(14,27,33,28)(16,30,35,36)(18,22,38,34)(19,26,20,39)(24,32,40,31);(2,4)(9,13)(10,18)(11,12)(14,32)(15,23)(16,26)(17,22)(19,36)(20,30)(21,38)(24,28)(25,29)(27,40)(31,33)(34,37)(35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326218);