#10312445

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

4594 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)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26,31,33)(27,30,36,32)(28,37,40,29)(34,35,38,39);(1,2,4,5)(3,6,7,8)(12,19)(16,22)(18,23)(20,24)(25,27)(26,32)(28,38)(29,39)(30,33)(31,36)(34,40)(35,37);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(16,22)(18,23)(25,26,31,33)(27,32,36,30)(28,37,40,29)(34,39,38,35);(2,5)(6,8)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,28,31,40)(26,29,33,37)(27,34,36,38)(30,39,32,35);(1,2)(3,8)(4,5)(6,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29)(26,28)(27,35)(30,38)(31,37)(32,34)(33,40)(36,39);(1,3)(2,6)(4,7)(5,8)(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,30,31,32)(26,27,33,36)(28,39,40,35)(29,34,37,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,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, 10219641);