#9684245

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

4149 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9591441);