#9054403

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,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, 8961599);