#6573995

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

2560 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)(7,19)(8,23)(9,16)(10,17)(11,14)(12,18)(13,21)(15,22)(20,32)(24,36)(25,29)(26,33)(27,30)(28,34)(31,35);(1,2,3,4)(5,7)(6,13)(8,12)(9,24)(10,21)(11,22)(14,23)(15,18)(16,31)(17,19)(20,27)(25,28)(26,35)(29,30)(32,34)(33,36);(6,14)(7,8)(11,12)(15,23)(16,30)(17,18)(20,24)(21,22)(25,35)(27,28)(31,36)(33,34);(5,8,10,22)(6,15,17,23)(7,12,21,11)(9,25,26,20)(13,18,19,14)(16,32,33,29)(24,28,35,27)(30,31,34,36);(5,9)(6,16)(7,20)(8,24)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(17,33)(18,34)(19,32)(21,25)(22,35)(23,36)

Polycyclic presentation:

2^9|b^2=fi,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6481191);