#9053978

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	#51775	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4202 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=g,e^2=i,f^2=g,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, 8961174);