#236577

order	512
nilpotency class	3
isoclinism	isoclinic class of #236557
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
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

1204 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)(4,6)(5,15)(7,9)(8,11)(12,16)(13,14)(17,18)(19,27)(20,25)(21,23)(22,26)(24,28)(29,32)(30,31);(1,3,6,12)(2,8,4,14)(5,13,7,11)(9,16,15,10)(18,23)(19,28)(20,21)(25,26)(27,31)(29,30);(1,4,6,2)(3,10,12,16)(5,15,7,9)(8,11,14,13)(17,19,22,30)(18,24,26,32)(20,28,21,29)(23,27,25,31);(17,20,22,21)(18,25,26,23)(19,29,30,28)(24,27,32,31)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gh,d^c=di,e^c=ei,f^c=fi,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, 143773);