#10250307

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

4378 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,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, 10157503);