#240728

order	512
nilpotency class	3
isoclinism	isoclinic class of #240721
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	#60958	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

868 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:

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

Polycyclic presentation:

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

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, 147924);