#10409775

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

4596 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10316971);