#174632

order	512
nilpotency class	3
isoclinism	isoclinic class of #174477
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

678 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)(15,23)(16,20)(18,21)(22,24)(25,26,29,33)(27,30,35,39)(28,31,36,32)(34,37,40,38);(3,4)(6,8)(9,10)(11,15)(12,20)(13,16)(14,19)(17,24)(18,22)(21,23)(25,27,29,35)(26,30,33,39)(28,34,36,40)(31,37,32,38);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,21)(13,17)(14,22)(16,24)(18,19)(20,23)(26,31)(28,36)(30,37)(32,33)(34,40)(38,39);(1,3,5,4)(2,6,7,8)(11,17)(15,24)(18,21)(22,23)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 81828);