#238918

order	512
nilpotency class	3
isoclinism	isoclinic class of #238917
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

880 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)(4,5)(6,10)(7,9)(8,13)(12,14)(15,16)(17,18)(19,26)(20,30)(21,24)(22,25)(23,28)(27,32)(29,31);(1,3,7,15)(2,8,9,12)(4,11,10,16)(5,14,6,13)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,32)(28,31);(1,4,5,2,7,10,6,9)(3,12,13,16,15,8,14,11)(17,19)(18,23)(20,29)(21,28)(22,27)(24,32)(25,31)(26,30);(2,9)(4,10)(8,12)(11,16)(19,27)(23,31)(26,32)(28,29)

Polycyclic presentation:

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

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, 146114);