#8904533

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4670 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8811729);