#238368

order	512
nilpotency class	3
isoclinism	isoclinic class of #238361
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

795 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)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18)(19,21)(20,25,28,23)(22,24)(26,30)(27,31,32,29);(4,5)(7,15)(8,14)(11,12)(17,19,21,18)(20,27)(22,26,30,24)(23,29)(25,31)(28,32);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,20,21,28)(18,23,19,25)(22,31,30,29)(24,27,26,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(20,28)(23,25)(27,32)(29,31)

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=i,d^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,f^b=fi,d^c=di,e^c=eh,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, 145564);