#6573918

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36009	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2639 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(2,4)(5,6)(7,8)(9,10,13,17)(11,19,23,15)(12,24,27,16)(14,18,29,36)(20,32,22,30)(21,31,25,33)(26,35,28,34)(37,39,38,40);(1,2,3,4)(9,11,13,23)(10,15,17,19)(12,25,27,21)(14,20,29,22)(16,33,24,31)(18,30,36,32)(26,38,28,37)(34,40,35,39);(6,8)(11,20)(12,26)(14,29)(15,30)(16,34)(18,36)(19,32)(22,23)(24,35)(27,28)(37,38)(39,40);(9,12,13,27)(10,16,17,24)(11,21,23,25)(14,26,29,28)(15,31,19,33)(18,34,36,35)(20,37,22,38)(30,39,32,40);(11,22)(12,26)(14,29)(15,32)(16,34)(18,36)(19,30)(20,23)(21,25)(24,35)(27,28)(31,33)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,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, 6481114);