#10407822

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

4894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=g,d^2=gi,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, 10315018);