#10324108

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

4477 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)(27,31)(28,35)(29,30)(32,36)(33,40)(34,38)(37,39);(1,2)(3,7)(4,5)(6,8)(9,11,15,13)(10,16,17,19)(12,20,23,21)(14,18,24,22)(26,30)(28,36)(31,38)(33,37);(2,5)(6,7)(10,17)(11,13)(18,22)(20,21)(25,27,29,34)(26,31,30,38)(28,33,36,37)(32,39,35,40);(2,5)(6,7)(9,12,15,23)(10,18,17,22)(11,21,13,20)(14,19,24,16)(25,28,29,36)(26,32,30,35)(27,33,34,37)(31,39,38,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,11)(10,19,17,16)(12,20,23,21)(14,18,24,22)(25,27)(26,31)(28,37)(29,34)(30,38)(32,40)(33,36)(35,39);(1,3)(2,6)(4,8)(5,7)(9,14,15,24)(10,20,17,21)(11,22,13,18)(12,19,23,16)(25,26)(27,31)(28,32)(29,30)(33,39)(34,38)(35,36)(37,40)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gh,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, 10231304);