#10529818

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	#58135	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4166 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)(3,8)(4,6)(5,7)(9,10,14,17)(11,13,16,19)(12,15,22,24)(18,21,20,23)(25,26,31,29)(27,34,30,32)(28,33,37,38)(35,40,36,39);(1,3,5,6)(2,4,7,8)(9,11)(10,13)(12,20)(14,16)(15,23)(17,19)(18,22)(21,24);(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,24)(19,23)(25,27,31,30)(26,32,29,34)(28,35,37,36)(33,39,38,40);(1,2)(3,8)(4,6)(5,7)(9,10)(11,19)(12,15)(13,16)(14,17)(18,23)(20,21)(22,24)(25,28)(26,33)(27,35)(29,38)(30,36)(31,37)(32,39)(34,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,15)(11,20)(13,23)(14,22)(16,18)(17,24)(19,21)(25,29,31,26)(27,34,30,32)(28,33,37,38)(35,39,36,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,19)(10,16,17,11)(12,21,22,23)(15,20,24,18)(25,30,31,27)(26,34,29,32)(28,36,37,35)(33,40,38,39)

Polycyclic presentation:

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

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, 10437014);