#8946235

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

4464 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8853431);