#178147

order	512
nilpotency class	3
isoclinism	isoclinic class of #177893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

693 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 85343);