#10312619

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

4354 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,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, 10219815);