#9065692

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	#35941	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4420 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=gh,f^2=gh,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, 8972888);