#9014316

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4222 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)(25,26)(27,33)(28,34)(29,32)(30,31)(35,39)(36,40)(37,38);(3,4)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(26,31)(27,34)(29,37)(36,39);(1,3)(2,7)(4,6)(5,8)(9,11,14,21)(10,15,18,19)(12,13,22,20)(16,24,23,17)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,40)(38,39);(3,4)(7,8)(9,12,14,22)(10,16,18,23)(11,20,21,13)(15,17,19,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40);(1,4,6,3)(2,7,5,8)(9,13,14,20)(10,17,18,24)(11,12,21,22)(15,23,19,16)(25,29,30,37)(26,32,31,38)(27,35,34,40)(28,36,33,39);(1,5,6,2)(3,7,4,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40)

Polycyclic presentation:

2^9|c^2=hi,d^2=hi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8921512);