#8925507

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

4672 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8832703);