#10511108

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

4513 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10418304);