#9625421

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	#51908	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4216 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)(11,20)(12,22)(13,17)(14,21)(15,16)(18,23)(19,24)(25,26,30,32)(27,33,35,28)(29,37,38,31)(34,36,40,39);(1,2)(3,4)(5,6)(7,8)(10,16)(11,13)(18,22)(19,21)(25,27,30,35)(26,28,32,33)(29,34,38,40)(31,36,37,39);(2,6)(3,7)(9,11,15,13)(10,17,16,20)(12,19,23,21)(14,18,24,22)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(3,7)(4,8)(9,12,15,23)(10,18,16,22)(11,21,13,19)(14,20,24,17)(25,29,30,38)(26,31,32,37)(27,34,35,40)(28,36,33,39);(1,3)(2,4)(5,7)(6,8)(9,13,15,11)(10,17,16,20)(12,19,23,21)(14,22,24,18);(1,4)(2,3)(5,8)(6,7)(9,14)(10,19)(11,18)(12,20)(13,22)(15,24)(16,21)(17,23)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9532617);