#10409684

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

4338 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

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, 10316880);