#9070719

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	#35681	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4566 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8977915);