#10219639

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

4191 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10126835);