#9053871

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	#56654	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4245 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)(11,21)(12,17)(13,18)(14,19)(15,20)(16,24)(22,23)(25,26)(27,35)(28,34)(29,33)(30,31)(32,36)(37,39)(38,40);(3,7)(6,8)(9,11,15,24)(10,16,20,21)(12,22,19,18)(13,14,23,17)(26,31)(27,36)(33,34)(37,38);(1,2)(3,6)(4,5)(7,8)(12,19)(13,23)(14,17)(18,22)(25,27)(26,32)(28,37)(29,38)(30,36)(31,35)(33,39)(34,40);(9,12,15,19)(10,17,20,14)(11,22,24,18)(13,21,23,16)(25,28,30,29)(26,33,31,34)(27,37,36,38)(32,39,35,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,23)(10,18,20,22)(11,17,24,14)(12,16,19,21)(25,26)(27,32)(28,33)(29,34)(30,31)(35,36)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,14,15,17)(10,19,20,12)(11,23,24,13)(16,18,21,22)(25,29,30,28)(26,34,31,33)(27,38,36,37)(32,40,35,39)

Polycyclic presentation:

2^9|b^2=gi,d^2=g,e^2=gi,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, 8961067);