#236702

order	512
nilpotency class	3
isoclinism	isoclinic class of #236695
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|d^2=hi,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gh,d^c=dhi,e^c=ei,f^c=fhi,g^c=gh

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, 143898);