#419638

order	512
nilpotency class	3
isoclinism	isoclinic class of #417704
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=ei,f^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,f^c=fi,g^c=gi,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, 326834);