#489034

order	512
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#447
prime radical	#33579	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=gh,d^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,h^a=hi,e^b=eg,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396230);