#10040859

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4035 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=ghi,e^2=h,f^2=h,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, 9948055);