#10103429

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	#51933	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4187 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10010625);