#9040955

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
prime radical	#35842	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4356 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8948151);