#9788855

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	#45554	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4168 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,e^2=g,f^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, 9696051);