#10047017

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

4475 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9954213);