#240715

order	512
nilpotency class	3
isoclinism	isoclinic class of #240712
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

894 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)(3,10)(4,11)(5,13)(6,9)(7,12)(8,15)(14,16)(17,18)(19,22)(20,23)(21,25)(24,29)(26,30)(27,32)(28,31);(1,3)(2,7)(4,8)(5,14)(6,11)(9,16)(10,13)(12,15)(17,19,21,27)(18,22,25,32)(20,28,29,26)(23,31,24,30);(1,4,5,7,6,10,15,16)(2,3,8,12,9,11,13,14)(18,23)(20,29)(22,30)(24,25)(26,28)(31,32);(3,11)(4,10)(7,16)(12,14)

Polycyclic presentation:

2^9|b^2=i,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=dh,f^c=fhi,g^c=gh

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, 147911);