#409645

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1154 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=hi,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316841);