#10247306

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

4789 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)(3,8)(4,6)(5,7)(9,10)(11,18)(12,21)(13,16)(14,15)(17,22)(19,23)(20,24)(25,26)(27,28)(29,37)(30,32)(31,39)(33,35)(34,38)(36,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,22)(21,24)(25,27)(26,28)(29,34)(30,35)(31,36)(32,33)(37,38)(39,40);(9,12,15,22)(10,17,14,21)(11,19,13,20)(16,23,18,24)(25,26,30,32)(27,33,35,28)(29,31,39,37)(34,38,40,36);(1,2,5,7)(3,8,4,6)(9,12)(10,17)(11,20)(13,19)(14,21)(15,22)(16,24)(18,23)(25,26)(27,33)(28,35)(29,31)(30,32)(34,38)(36,40)(37,39);(1,2)(3,6)(4,8)(5,7)(9,13,15,11)(10,16,14,18)(12,20,22,19)(17,23,21,24)(25,28,30,33)(26,27,32,35)(29,38,39,36)(31,40,37,34);(1,4,5,3)(2,6,7,8)(9,14,15,10)(11,18,13,16)(12,21,22,17)(19,24,20,23)(25,29)(26,31)(27,34)(28,36)(30,39)(32,37)(33,38)(35,40)

Polycyclic presentation:

2^9|b^2=ghi,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, 10154502);