#269502

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	#59034	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1045 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=hi,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, 176698);