#9040988

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	#51896	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4572 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8948184);