#9210364

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	#38433	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4274 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9117560);