#179889

order	512
nilpotency class	3
isoclinism	isoclinic class of #179429
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

840 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,d^b=dgi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87085);