#6439024

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

3091 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)(7,14)(8,22)(9,15)(10,16)(11,17)(12,19)(13,25)(18,27)(20,29)(21,28)(23,30)(24,32)(26,31)(33,35)(34,36);(5,7,9,19,11,21,10,20)(6,12,16,14,17,29,15,28)(8,23,24,18,26,33,25,34)(13,30,31,36,32,35,22,27);(2,4)(5,8,9,24,11,26,10,25)(6,13,16,31,17,32,15,22)(7,18,19,33,21,34,20,23)(12,27,14,30,29,36,28,35);(5,7)(6,14)(8,18)(9,19)(10,20)(11,21)(12,15)(13,30)(16,29)(17,28)(22,27)(23,25)(24,33)(26,34)(31,36)(32,35);(5,9,11,10)(6,15,17,16)(7,19,21,20)(8,24,26,25)(12,28,29,14)(13,22,32,31)(18,33,34,23)(27,35,36,30)

Polycyclic presentation:

2^9|b^2=f,c^2=f,e^2=i,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=dhi,h^b=hi,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, 6346220);