#10040632

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4091 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9947828);