#8908301

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

4481 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)(3,4)(5,6)(7,8)(9,10,14,18)(11,15,19,22)(12,16,21,24)(13,17,23,20)(25,26)(27,35)(28,31)(29,33)(30,38)(32,36)(34,40)(37,39);(2,6)(3,7)(9,11)(10,15)(12,20)(13,16)(14,19)(17,21)(18,22)(23,24)(25,27,31,36)(26,32,28,35)(29,38,39,34)(30,33,40,37);(1,3,5,7)(2,4,6,8)(11,19)(12,21)(15,22)(16,24)(26,28)(30,40)(32,35)(33,37);(1,4)(2,7)(3,6)(5,8)(9,12)(10,16)(11,20)(13,15)(14,21)(17,19)(18,24)(22,23)(25,28,31,26)(27,35,36,32)(29,33,39,37)(30,38,40,34);(9,13)(10,17)(11,16)(12,22)(14,23)(15,21)(18,20)(19,24)(25,29,31,39)(26,33,28,37)(27,34,36,38)(30,35,40,32);(9,11)(10,15)(12,20)(13,16)(14,19)(17,21)(18,22)(23,24)(25,30,31,40)(26,34,28,38)(27,33,36,37)(29,35,39,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=h,e^2=h,f^2=h,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, 8815497);