#10320908

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

4498 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:

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

Polycyclic presentation:

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

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, 10228104);