#9607693

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4069 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)(11,15)(12,16)(13,24)(14,18)(17,21)(19,20)(22,23)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,2,5,4)(3,7,8,6)(9,11,14,20)(10,15,18,19)(12,21,23,13)(16,17,22,24)(26,27)(31,33)(32,34)(38,39);(1,3,5,8)(2,6,4,7)(9,10,14,18)(11,19,20,15)(12,22,23,16)(13,24,21,17)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,3,5,8)(2,7,4,6)(9,10)(11,15)(12,22)(13,17)(14,18)(16,23)(19,20)(21,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,13)(14,23)(15,24)(17,19)(18,22)(20,21)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39);(1,4,5,2)(3,6,8,7)(9,13)(10,17)(11,12)(14,21)(15,22)(16,19)(18,24)(20,23)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=hi,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9514889);