#9101907

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4215 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,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, 9009103);