#6439030

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439029
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346226);