#9945582

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4527 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9852778);