#9684145

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	#37777	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4396 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)(11,20)(12,21)(13,19)(14,16)(15,17)(18,22)(23,24)(26,31)(32,33)(34,38)(39,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,14)(10,16,17,20)(12,19,22,23)(13,21,24,18)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,3,5,6)(2,4,7,8)(10,17)(12,22)(16,20)(19,23)(25,27,30,28)(26,32,31,33)(29,35,37,36)(34,39,38,40);(9,12)(10,18)(11,19)(13,16)(14,23)(15,22)(17,21)(20,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,2)(3,4)(5,7)(6,8)(9,13)(10,19)(11,18)(12,16)(14,21)(15,24)(17,23)(20,22)(25,28,30,27)(26,33,31,32)(29,35,37,36)(34,39,38,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,20,17,16)(12,23,22,19)(13,18,24,21)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,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=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, 9591341);