#10089056

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4215 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)(11,16)(12,13)(14,22)(15,17)(18,24)(19,21)(20,23)(25,26,32,37)(27,28,39,33)(29,40,38,34)(30,35,36,31);(1,2)(3,6)(4,7)(5,8)(9,11,15,12)(10,16,17,13)(14,19,24,20)(18,23,22,21)(25,27)(26,33)(28,37)(29,35)(30,40)(31,38)(32,39)(34,36);(1,3)(2,5)(4,8)(6,7)(10,17)(11,12)(18,22)(19,20)(25,28,32,33)(26,27,37,39)(29,36,38,30)(31,40,35,34);(1,2)(3,5)(4,7)(6,8)(9,12,15,11)(10,13,17,16)(14,20,24,19)(18,21,22,23)(25,29,32,38)(26,34,37,40)(27,35,39,31)(28,30,33,36);(1,3)(2,6)(4,8)(5,7)(9,13)(10,12)(11,17)(14,23)(15,16)(18,19)(20,22)(21,24)(25,30)(26,35)(27,34)(28,29)(31,37)(32,36)(33,38)(39,40);(1,2)(3,5)(4,7)(6,8)(9,14,15,24)(10,18,17,22)(11,19,12,20)(13,21,16,23)(25,31,32,35)(26,36,37,30)(27,38,39,29)(28,40,33,34)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=h,f^2=h,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, 9996252);