#10349119

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

4400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=hi,d^2=gh,e^2=gh,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, 10256315);