#239853

order	512
nilpotency class	3
isoclinism	isoclinic class of #234006
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1036 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=eh,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dhi,f^b=fi,g^b=gi,d^c=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147049);