#236597

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
prime radical	#60916	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1192 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)(3,11)(4,8)(5,9)(6,10)(7,14)(12,15)(13,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,30)(28,31)(29,32);(1,3)(2,7)(4,13)(5,14)(6,15)(8,11)(9,12)(10,16)(17,19,22,29)(18,23,26,31)(20,28,21,30)(24,32,25,27);(1,2)(3,12)(4,9)(5,8)(6,10)(7,14)(11,15)(13,16)(18,24)(19,28)(20,21)(25,26)(27,32)(29,30);(1,4,6,5)(2,8,10,9)(3,13,15,14)(7,11,16,12)(19,29)(23,31)(27,32)(28,30)

Polycyclic presentation:

2^9|b^2=i,d^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,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, 143793);