#10407805

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

4421 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=hi,e^2=hi,f^2=hi,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, 10315001);