#269587

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
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

1021 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=hi,c^2=i,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=deh,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,f^c=fhi,e^d=ei,f^d=fhi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 176783);