#239000

order	512
nilpotency class	3
isoclinism	isoclinic class of #239000
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×#3253	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

960 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)(3,13)(4,8)(5,11)(6,10)(7,12)(9,15)(14,16)(17,18)(19,27)(20,23)(21,25)(22,26)(24,30)(28,32)(29,31);(1,3)(2,8)(4,6)(5,15)(7,14)(9,12)(10,16)(11,13)(18,23)(19,28)(20,21)(24,27)(25,26)(31,32);(1,4,5,14,7,15,6,3)(2,9,10,16,12,8,11,13)(17,19,20,29,22,28,21,30)(18,24,25,32,26,31,23,27);(19,28)(24,31)(27,32)(29,30)

Polycyclic presentation:

2^9|c^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,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, 146196);