#236560

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	#60968	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1306 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,14)(15,16)(19,25)(20,27)(22,30)(24,28)(26,29)(31,32);(1,3,7,15)(2,8,12,4)(5,14,6,9)(10,16,11,13)(17,18)(19,22)(20,28)(21,23)(24,27)(25,32)(26,31)(29,30);(1,4)(2,9)(3,10)(5,13)(6,16)(7,8)(11,15)(12,14)(17,19)(18,22)(20,29)(21,26)(23,31)(24,32)(25,27)(28,30);(1,5,7,6)(2,10,12,11)(3,14,15,9)(4,13,8,16)(18,23)(19,26)(24,28)(25,29)

Polycyclic presentation:

2^9|b^2=h,d^2=h,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, 143756);