#241012

order	512
nilpotency class	3
isoclinism	isoclinic class of #241009
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

914 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)(3,11)(4,6)(5,9)(7,10)(8,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,25)(24,29)(26,31)(27,30)(28,32);(1,3,7,15)(2,8,10,16)(4,11,9,12)(5,14,6,13)(17,19)(18,22)(20,26)(21,27)(23,31)(24,32)(25,30)(28,29);(1,4,6,2,7,9,5,10)(3,12,14,8,15,11,13,16)(18,23)(19,26)(20,29)(24,25)(27,28)(31,32);(1,5,7,6)(2,9,10,4)(3,13,15,14)(8,11,16,12)(19,27)(22,30)(26,28)(31,32)

Polycyclic presentation:

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

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, 148208);