#236591

order	512
nilpotency class	3
isoclinism	isoclinic class of #236579
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

1407 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)(3,7)(4,6)(5,9)(8,13)(10,16)(11,15)(12,14)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,31)(28,32);(2,6)(3,10)(4,13)(8,9)(11,12)(14,16)(20,21)(24,30)(26,32)(27,29);(1,3)(2,7)(4,12)(5,11)(6,14)(8,16)(9,15)(10,13)(17,19)(18,23)(20,29)(21,27)(22,28)(24,32)(25,31)(26,30);(3,11)(7,15)(10,12)(14,16)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,26,31,32)

Polycyclic presentation:

2^9|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=dhi,e^c=ei,f^c=fi,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, 143787);