#9063939

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4239 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,e^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, 8971135);