#179672

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×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

799 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)(11,19)(12,16)(13,22)(14,17)(15,20)(18,23)(21,24)(25,26,30,32)(27,34,31,39)(28,36,37,35)(29,38,40,33);(1,2)(3,5)(4,6)(7,8)(9,11,14,12)(10,15,17,24)(13,21,23,20)(16,22,19,18)(25,27,30,31)(26,28,32,37)(29,36,40,35)(33,39,38,34);(1,3)(2,5)(4,7)(6,8)(9,12,14,11)(10,16,17,19)(13,21,23,20)(15,22,24,18)(25,28)(26,31)(27,32)(29,39)(30,37)(33,35)(34,40)(36,38);(10,17)(11,12)(18,22)(20,21)(26,32)(28,37)(33,38)(34,39)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^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, 86868);