#6573996

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

2558 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)(7,16)(8,22)(9,13)(10,15)(11,19)(12,26)(14,27)(17,32)(18,21)(20,30)(23,29)(24,35)(25,34)(28,33)(31,36);(1,2,3,4)(5,7,9,19)(6,11,13,16)(8,23,25,24)(10,20,27,21)(12,31,33,32)(14,30,15,18)(17,28,36,26)(22,35,34,29);(5,8,9,25)(6,12,13,33)(7,17,19,36)(10,28,27,26)(11,29,16,35)(14,34,15,22)(18,31,30,32)(20,23,21,24);(5,7,9,19)(6,11,13,16)(8,24,25,23)(10,20,27,21)(12,32,33,31)(14,30,15,18)(17,26,36,28)(22,29,34,35);(5,6)(7,18)(8,12)(9,13)(10,14)(11,20)(15,27)(16,21)(17,31)(19,30)(22,26)(23,29)(24,35)(25,33)(28,34)(32,36)

Polycyclic presentation:

2^9|b^2=f,c^2=i,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, 6481192);