#10089285

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	#47844	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4530 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9996481);