#10153072

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

4410 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,e^2=gi,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, 10060268);