#450126

order	512
nilpotency class	3
isoclinism	isoclinic class of #418488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1053 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ef,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^b=hi,f^c=fi,g^c=gi

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, 357322);