#10529244

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

4395 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)(3,4)(5,6)(7,8)(9,10,14,17)(11,12,19,15)(13,21,23,16)(18,24,22,20)(25,26,30,33)(27,29,32,35)(28,31,38,40)(34,37,36,39);(2,6)(3,7)(9,10,14,17)(11,12,19,15)(13,16,23,21)(18,20,22,24)(25,27)(26,29)(28,36)(30,32)(31,39)(33,35)(34,38)(37,40);(1,3,5,7)(2,4,6,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,24)(20,21)(22,23)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,40)(35,39);(1,4)(2,7)(3,6)(5,8)(9,12,14,15)(10,11,17,19)(13,20,23,24)(16,18,21,22)(25,26)(27,35)(28,31)(29,32)(30,33)(34,39)(36,37)(38,40);(9,11)(10,12)(13,22)(14,19)(15,17)(16,24)(18,23)(20,21)(25,28)(26,31)(27,36)(29,39)(30,38)(32,34)(33,40)(35,37);(9,13,14,23)(10,16,17,21)(11,18,19,22)(12,20,15,24)(25,29,30,35)(26,32,33,27)(28,37,38,39)(31,36,40,34)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=h,f^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, 10436440);