#9125975

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	#42308	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4319 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9033171);