#9779499

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	#47856	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4249 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gh,e^2=g,f^2=g,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, 9686695);