#10145964

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

4419 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=gi,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, 10053160);