#10480838

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

4725 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10388034);