#10247305

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

4570 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:

(1,2,5,4)(3,7,8,6)(9,10)(11,12)(13,21)(14,16)(15,23)(17,19)(18,22)(20,24)(25,26)(27,33)(28,37)(29,31)(30,34)(32,35)(36,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,18)(14,19)(15,20)(16,17)(21,22)(23,24)(25,27,32,29)(26,33,35,31)(28,36,30,39)(34,40,37,38);(9,10,14,16)(11,17,19,12)(13,15,23,21)(18,22,24,20)(25,28,32,30)(26,34,35,37)(27,36,29,39)(31,38,33,40);(1,3,5,8)(2,6,4,7)(9,10)(11,17)(12,19)(13,15)(14,16)(18,22)(20,24)(21,23)(25,29,32,27)(26,31,35,33)(28,36,30,39)(34,40,37,38);(1,3)(2,7)(4,6)(5,8)(9,12,14,17)(10,11,16,19)(13,22,23,20)(15,24,21,18)(25,30,32,28)(26,34,35,37)(27,36,29,39)(31,40,33,38);(1,4,5,2)(3,7,8,6)(9,13)(10,15)(11,18)(12,20)(14,23)(16,21)(17,22)(19,24)(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=hi,c^2=h,d^2=ghi,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, 10154501);