#9937823

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

4498 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)(4,6)(5,7)(9,10)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,26,30,34)(27,35,36,31)(28,32,33,29)(37,38,40,39);(1,3)(2,6)(4,5)(7,8)(11,17)(14,21)(18,23)(22,24)(27,36)(31,35)(37,40)(38,39);(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,27,30,36)(26,31,34,35)(28,37,33,40)(29,38,32,39);(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,27)(26,31)(28,37)(29,38)(30,36)(32,39)(33,40)(34,35);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,28)(26,32)(27,37)(29,34)(30,33)(31,39)(35,38)(36,40);(1,2)(3,6)(4,8)(5,7)(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,29,30,32)(26,33,34,28)(27,38,36,39)(31,40,35,37)

Polycyclic presentation:

2^9|a^2=i,c^2=h,e^2=g,f^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, 9845019);