#240731

order	512
nilpotency class	3
isoclinism	isoclinic class of #240730
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

905 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,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=dhi,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, 147927);