#6573823

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2643 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481019);