#6573831

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

2671 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)(11,19,22,15)(12,24,27,16)(14,18,28,34)(20,31,23,29)(21,30,37,35)(25,33,26,32)(36,39,38,40);(1,2,3,4)(9,11,13,22)(10,15,17,19)(12,21,27,37)(14,20,28,23)(16,30,24,35)(18,29,34,31)(25,38,26,36)(32,40,33,39);(6,8)(11,20)(12,25)(14,28)(15,29)(16,32)(18,34)(19,31)(21,37)(22,23)(24,33)(26,27)(30,35);(9,12,13,27)(10,16,17,24)(11,21,22,37)(14,26,28,25)(15,30,19,35)(18,33,34,32)(20,36,23,38)(29,39,31,40);(11,20)(12,26)(14,28)(15,29)(16,33)(18,34)(19,31)(22,23)(24,32)(25,27)(36,38)(39,40)

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,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, 6481027);