#10324103

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4227 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)(11,16)(12,18)(13,19)(14,21)(15,17)(20,24)(22,23)(25,26,31,32)(27,35,39,33)(28,34,30,29)(36,40,38,37);(1,2)(3,7)(4,5)(6,8)(9,11,15,13)(10,16,17,19)(12,20,23,21)(14,18,24,22)(26,32)(28,30)(33,35)(36,38);(2,5)(6,7)(10,17)(11,13)(18,22)(20,21)(25,27,31,39)(26,33,32,35)(28,36,30,38)(29,40,34,37);(2,5)(6,7)(9,12,15,23)(10,18,17,22)(11,21,13,20)(14,19,24,16)(25,28,31,30)(26,29,32,34)(27,36,39,38)(33,40,35,37);(1,2)(3,6)(4,5)(7,8)(9,13,15,11)(10,19,17,16)(12,20,23,21)(14,18,24,22)(25,29,31,34)(26,28,32,30)(27,37,39,40)(33,38,35,36);(1,3)(2,6)(4,8)(5,7)(9,14,15,24)(10,20,17,21)(11,22,13,18)(12,19,23,16)(25,30,31,28)(26,34,32,29)(27,38,39,36)(33,37,35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=h,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, 10231299);