#9783183

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	#59163	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4256 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9690379);