#172152

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

817 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,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, 79348);