#6443312

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437931
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,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, 6350508);