#6443310

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

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

5044 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6350506);