#10323120

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

4335 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)(12,20)(13,15)(14,19)(16,22)(18,23)(21,24)(25,26)(27,32)(28,29)(30,39)(31,35)(33,38)(34,36)(37,40);(1,2)(3,7)(4,5)(6,8)(9,11,13,19)(10,14,15,17)(12,21,22,18)(16,24,20,23)(25,27,31,37)(26,32,35,40)(28,36,38,30)(29,34,33,39);(2,5)(6,7)(10,15)(14,17)(16,20)(23,24)(25,28)(26,33)(27,36)(29,35)(30,37)(31,38)(32,39)(34,40);(2,5)(6,7)(9,12,13,22)(10,16,15,20)(11,18,19,21)(14,23,17,24)(25,29,31,33)(26,28,35,38)(27,34,37,39)(30,32,36,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,23)(20,22)(21,24)(25,28)(26,29)(27,36)(30,37)(31,38)(32,34)(33,35)(39,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,24)(21,22)(25,30,31,36)(26,34,35,39)(27,28,37,38)(29,32,33,40)

Polycyclic presentation:

2^9|b^2=h,d^2=h,f^2=gh,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, 10230316);