#10097048

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4139 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)(11,16)(12,13)(14,21)(15,17)(18,20)(19,23)(22,24)(25,26,31,34)(27,33,38,32)(28,30,29,39)(35,40,36,37);(1,2,5,8)(3,4,7,6)(9,11)(10,16)(12,20)(13,18)(14,22)(15,19)(17,23)(21,24)(25,27)(26,32)(28,35)(29,36)(30,37)(31,38)(33,34)(39,40);(1,3)(2,6)(4,8)(5,7)(10,17)(11,19)(12,22)(18,21)(25,28,31,29)(26,30,34,39)(27,35,38,36)(32,37,33,40);(1,4)(2,7)(3,8)(5,6)(9,12,15,22)(10,13,17,24)(11,20,19,14)(16,18,23,21)(25,29,31,28)(26,30,34,39)(27,36,38,35)(32,37,33,40);(1,3)(2,4)(5,7)(6,8)(9,13)(10,12)(11,21)(14,16)(15,24)(17,22)(18,19)(20,23)(25,30)(26,29)(27,37)(28,34)(31,39)(32,36)(33,35)(38,40);(1,4)(2,7)(3,8)(5,6)(9,14,15,20)(10,18,17,21)(11,22,19,12)(13,23,24,16)(25,27)(26,33)(28,35)(29,36)(30,40)(31,38)(32,34)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=gh,f^2=h,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, 10004244);