#9683429

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	#37827	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4208 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:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=h,e^2=g,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, 9590625);