#236074

order	512
nilpotency class	3
isoclinism	isoclinic class of #234044
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,9,13)(7,11,14,18)(8,10,16,12)(15,17,20,19)(22,26)(23,29)(24,32)(28,33)(31,34)(35,36);(1,2)(3,4)(6,10)(8,16)(11,17)(12,13)(15,20)(18,19)(21,22)(23,30)(24,33)(25,28)(26,32)(27,31)(29,36)(34,35);(2,4)(5,7,9,14)(6,11,13,18)(8,15,16,20)(10,17,12,19)(21,23,25,31)(22,27,28,30)(24,34,32,29)(26,35,33,36);(7,14)(11,18)(15,20)(17,19)(22,28)(26,33)(27,30)(35,36)

Polycyclic presentation:

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

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, 143270);