#236500

order	512
nilpotency class	3
isoclinism	isoclinic class of #236489
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	#60968	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1325 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:

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

Polycyclic presentation:

2^9|b^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,e^b=eh,f^b=fi,g^b=gh,d^c=dhi,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, 143696);