#8933833

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	#59184	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4461 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8841029);