#6583187

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574063
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2994 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)(7,15)(8,13)(9,14)(10,17)(11,18)(12,20)(16,22)(19,34)(21,29)(23,30)(24,31)(25,35)(26,32)(27,33)(28,36);(5,7,10,20,11,8,22,21)(6,12,16,15,18,29,17,13)(9,19,25,36,27,23,26,24)(14,28,32,34,33,31,35,30);(1,2,3,4)(5,8)(6,13)(7,11)(9,24)(10,21)(12,16)(14,31)(15,18)(17,29)(19,25)(20,22)(23,26)(27,36)(28,33)(30,32)(34,35);(5,9)(6,14)(7,19)(8,23)(10,25)(11,27)(12,28)(13,30)(15,34)(16,32)(17,35)(18,33)(20,36)(21,24)(22,26)(29,31);(5,7)(6,15)(8,11)(9,24)(10,20)(12,17)(13,18)(14,31)(16,29)(19,25)(21,22)(23,26)(27,36)(28,33)(30,32)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490383);