#10368651

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

4481 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=i,d^2=hi,e^2=g,f^2=g,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, 10275847);