#9947041

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

4581 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=gi,e^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, 9854237);