#6443032

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437793
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2506 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)(5,6,9,24,11,15,10,14)(7,16,25,12,19,27,18,28)(8,13,21,31,23,32,22,30)(17,33,20,26,35,29,34,36);(5,7)(6,12)(8,20)(9,18)(10,25)(11,19)(13,29)(14,27)(15,28)(16,24)(17,21)(22,35)(23,34)(26,31)(30,36)(32,33);(1,2,3,4)(5,8)(6,13)(7,17)(9,21)(10,22)(11,23)(12,26)(14,30)(15,32)(16,33)(18,34)(19,35)(20,25)(24,31)(27,29)(28,36);(5,8)(6,13)(7,17)(9,21)(10,22)(11,23)(12,26)(14,30)(15,32)(16,33)(18,34)(19,35)(20,25)(24,31)(27,29)(28,36);(5,9,11,10)(6,14,15,24)(7,18,19,25)(8,21,23,22)(12,27,28,16)(13,30,32,31)(17,34,35,20)(26,29,36,33)

Polycyclic presentation:

2^9|a^2=fi,c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,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, 6350228);