#424737

order	512
nilpotency class	3
isoclinism	isoclinic class of #419794
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1140 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:

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

Polycyclic presentation:

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

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, 331933);