#446278

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

850 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)(11,18)(12,26)(13,28,31,37)(14,21)(15,20)(16,22)(17,24)(19,30,36,29)(23,38,39,32)(25,33)(27,35,40,34);(1,3,5,8)(2,6,4,7)(9,11)(10,17)(12,14)(13,27)(15,24)(16,25)(18,21)(19,35)(20,33)(22,26)(23,29)(28,32)(30,39)(31,40)(34,36)(37,38);(1,2,5,4)(3,6,8,7)(9,12,14,11,16,24,15,25)(10,18,20,26,22,33,21,17)(13,27,30,23,31,40,29,39)(19,34,37,38,36,35,28,32);(1,4,5,2)(3,6,8,7)(9,13,16,31)(10,19,22,36)(11,23,25,39)(12,27,24,40)(14,30,15,29)(17,32,26,38)(18,34,33,35)(20,37,21,28)

Polycyclic presentation:

2^9|a^2=eg,b^2=efi,c^2=efg,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei,f^d=fi,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, 353474);