#10352419

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
prime radical	#36724	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4358 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)(3,8)(4,7)(5,6)(9,10,14,12)(11,15,17,19)(13,16,22,20)(18,23,21,24)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,36)(39,40);(1,3,6,4)(2,7,5,8)(9,11)(10,15)(12,19)(13,21)(14,17)(16,24)(18,22)(20,23);(1,2,6,5)(3,7,4,8)(11,17)(15,19)(18,21)(23,24)(25,27,30,31)(26,28,33,34)(29,35,38,39)(32,36,37,40);(1,2,6,5)(3,8,4,7)(9,12,14,10)(11,15,17,19)(13,20,22,16)(18,23,21,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,38)(39,40);(1,4,6,3)(2,7,5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,24)(21,22)(25,28)(26,31)(27,33)(29,36)(30,34)(32,39)(35,37)(38,40);(1,5,6,2)(3,7,4,8)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=gi,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, 10259615);