#172226

order	512
nilpotency class	3
isoclinism	isoclinic class of #172137
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

713 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=h,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, 79422);