#10247599

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

4500 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154795);