#9781187

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	#54921	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4372 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,f^2=gh,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, 9688383);