#240913

order	512
nilpotency class	3
isoclinism	isoclinic class of #240913
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	#60803	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

965 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:

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

Polycyclic presentation:

2^9|c^2=e,e^2=h,f^2=i,g^2=h,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,g^b=gh,d^c=di,f^c=fhi

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, 148109);