#9133767

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	#37694	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4646 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)(9,10,15,19)(11,16,20,24)(12,17,21,22)(13,14,18,23)(25,26,30,31)(27,28,33,32)(29,37,39,34)(35,38,40,36);(3,7)(5,8)(9,11,15,20)(10,16,19,24)(12,13,21,18)(14,22,23,17)(26,31)(28,32)(34,37)(36,38);(1,2)(3,5)(4,6)(7,8)(11,20)(12,21)(16,24)(17,22)(25,27,30,33)(26,32,31,28)(29,35,39,40)(34,38,37,36);(9,12)(10,17)(11,13)(14,16)(15,21)(18,20)(19,22)(23,24)(25,27)(26,32)(28,31)(29,35)(30,33)(34,38)(36,37)(39,40);(1,3)(2,5)(4,7)(6,8)(9,13)(10,14)(11,21)(12,20)(15,18)(16,22)(17,24)(19,23)(25,28,30,32)(26,33,31,27)(29,38,39,36)(34,35,37,40);(1,2)(3,5)(4,6)(7,8)(9,14)(10,18)(11,22)(12,16)(13,19)(15,23)(17,20)(21,24)(25,29,30,39)(26,34,31,37)(27,35,33,40)(28,36,32,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,e^2=g,f^2=g,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, 9040963);