#242100

order	512
nilpotency class	3
isoclinism	isoclinic class of #234044
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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

750 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)(3,6)(4,5)(7,8)(9,10,12,16)(11,18,19,14)(13,15,21,17)(20,23,24,22)(26,31)(27,35)(29,30)(32,36)(37,38)(39,40);(2,5)(6,8)(9,11,12,19)(10,14,16,18)(13,20,21,24)(15,22,17,23)(25,26,28,31)(27,36,37,40)(29,34,30,33)(32,35,39,38);(10,15)(13,21)(14,22)(16,17)(18,23)(20,24)(25,27,28,37)(26,32,31,39)(29,38,30,35)(33,36,34,40);(1,3,4,7)(2,6,5,8)(11,19)(14,18)(20,24)(22,23)(26,31)(27,37)(33,34)(35,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149296);