#9053865

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	#52417	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961061);