#9947408

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

4432 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=i,e^2=h,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, 9854604);