#239026

order	512
nilpotency class	3
isoclinism	isoclinic class of #239025
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

837 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)(4,8)(5,15)(6,9)(7,12)(10,14)(13,16)(17,18)(19,27)(20,21)(22,26)(23,25)(24,29)(28,31)(30,32);(1,3,6,13)(2,7,9,8)(4,14,12,5)(10,16,15,11)(17,19,23,31)(18,24,25,32)(20,27,26,28)(21,30,22,29);(1,4,5,3,6,12,14,13)(2,8,10,16,9,7,15,11)(17,20,21,18,23,26,22,25)(19,28,29,24,31,27,30,32);(2,9)(7,8)(10,15)(11,16)(18,25)(20,26)(24,32)(27,28)

Polycyclic presentation:

2^9|b^2=h,c^2=e,e^2=h,f^2=i,g^2=hi,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=ghi,d^c=di,f^c=fhi,g^c=gi

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, 146222);