#9794287

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

4323 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)(3,8)(4,7)(5,6)(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26,30,34)(27,31,36,37)(28,32,38,40)(29,33,39,35);(2,6)(7,8)(9,11)(10,14)(12,20)(13,19)(15,23)(16,17)(18,21)(22,24)(25,27,30,36)(26,31,34,37)(28,33,38,35)(29,32,39,40);(9,11,13,19)(10,14,16,17)(12,18,21,20)(15,22,24,23)(25,27)(26,31)(28,35)(29,32)(30,36)(33,38)(34,37)(39,40);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,21)(23,24);(1,2)(3,7)(4,8)(5,6)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,32)(27,35)(29,31)(30,38)(33,36)(34,40)(37,39);(1,4,5,3)(2,7,6,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,29)(26,33)(27,32)(28,37)(30,39)(31,38)(34,35)(36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9701483);