#9674921

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

4293 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9582117);