#10322264

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	#47649	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4166 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gh,e^2=gh,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, 10229460);