#238974

order	512
nilpotency class	3
isoclinism	isoclinic class of #238970
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	#60968	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

928 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)(3,7)(4,6)(5,8)(9,12)(10,13)(11,14)(15,16)(18,23)(19,26)(20,21)(24,31)(25,29)(27,28);(2,6)(3,10)(4,12)(7,14)(8,9)(11,16)(17,18)(19,24)(20,23)(21,29)(22,25)(26,30)(27,32)(28,31);(1,3,4,10,5,11,12,16)(2,7,9,15,8,14,6,13)(17,19,21,27,22,28,20,26)(18,24,23,30,25,31,29,32);(2,8)(6,9)(7,14)(13,15)(17,20,22,21)(18,23,25,29)(19,27,28,26)(24,32,31,30)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146170);