#10145807

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4200 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^2=i,f^2=gh,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, 10053003);