#10040496

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
prime radical	#47679	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4283 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)(3,6)(4,7)(5,8)(9,10,15,21)(11,22,23,16)(12,18,24,17)(13,19,20,14)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,14)(13,17)(15,23)(18,20)(19,24)(21,22)(27,34)(28,37)(31,33)(32,36);(1,3)(2,6)(4,8)(5,7)(9,12,15,24)(10,17,21,18)(11,14,23,19)(13,22,20,16)(25,27,30,34)(26,31,29,33)(28,38,37,35)(32,40,36,39);(1,2)(3,6)(4,7)(5,8)(9,12)(10,18)(11,19)(13,16)(14,23)(15,24)(17,21)(20,22)(25,28,30,37)(26,32,29,36)(27,35,34,38)(31,39,33,40);(1,2)(3,5)(4,7)(6,8)(9,13)(10,19)(11,18)(12,16)(14,21)(15,20)(17,23)(22,24)(25,27)(26,33)(28,35)(29,31)(30,34)(32,40)(36,39)(37,38);(1,3)(2,6)(4,8)(5,7)(9,14,15,19)(10,20,21,13)(11,12,23,24)(16,18,22,17)(25,29,30,26)(27,33,34,31)(28,36,37,32)(35,40,38,39)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9947692);