#8772588

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

4278 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)(11,19)(12,22)(13,23)(14,18)(15,21)(16,20)(17,24)(25,26,30,31)(27,33,36,32)(28,29,37,38)(34,39,40,35);(1,3,6,5)(2,7,4,8)(9,11)(10,15)(12,13)(14,21)(16,23)(17,22)(18,19)(20,24)(26,31)(28,37)(32,33)(34,40);(1,3,6,5)(2,8,4,7)(9,10,14,18)(11,15,21,19)(12,22,20,16)(13,17,24,23)(25,26)(27,32)(28,29)(30,31)(33,36)(34,35)(37,38)(39,40);(9,12,14,20)(10,16,18,22)(11,13,21,24)(15,23,19,17)(25,27,30,36)(26,32,31,33)(28,34,37,40)(29,35,38,39);(1,4,6,2)(3,7,5,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,22)(23,24)(25,28)(26,29)(27,34)(30,37)(31,38)(32,35)(33,39)(36,40);(1,5,6,3)(2,7,4,8)(9,13)(10,17)(11,20)(12,21)(14,24)(15,16)(18,23)(19,22)(25,29)(26,28)(27,35)(30,38)(31,37)(32,34)(33,40)(36,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,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,f^b=fgi,d^c=dgi,e^c=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, 8679784);