#9523993

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	C₂×#538	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4131 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=h,e^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, 9431189);