#172475

order	512
nilpotency class	3
isoclinism	isoclinic class of #172137
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

859 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)(10,14)(12,19)(15,21)(16,20)(18,23)(22,24)(25,26,30,33)(27,31,29,37)(28,35,36,39)(32,38,40,34);(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,15,17,22)(12,20,19,14)(18,24,23,21)(25,27,30,29)(26,31,33,37)(28,34,36,38)(32,39,40,35);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,23)(20,24)(25,28,30,36)(26,32,33,40)(27,34,29,38)(31,39,37,35);(2,6)(5,8)(10,16)(11,17)(14,20)(18,23)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 79671);