#240619

order	512
nilpotency class	3
isoclinism	isoclinic class of #240611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60968	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

945 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=e,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,f^c=fhi,g^c=gh

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, 147815);