#10042554

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

4614 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)(3,4)(5,6)(7,8)(9,10)(11,13)(12,15)(14,20)(16,18)(17,22)(19,23)(21,24)(25,26)(27,34)(28,33)(29,31)(30,32)(35,40)(36,38)(37,39);(1,3,6,7)(2,4,5,8)(9,11)(10,16)(12,13)(14,19)(15,18)(17,23)(20,21)(22,24)(25,27,30,29)(26,31,32,34)(28,36,37,35)(33,40,39,38);(1,4,6,8)(2,7,5,3)(10,12)(11,18)(17,20)(19,24)(26,32)(28,37)(31,34)(35,36);(1,2,6,5)(3,8,7,4)(9,12,15,10)(11,13,18,16)(14,20,22,17)(19,21,24,23)(25,28)(26,33)(27,35)(29,36)(30,37)(31,38)(32,39)(34,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,11)(12,18)(14,21)(15,16)(17,19)(20,24)(22,23)(25,26)(27,34)(28,33)(29,31)(30,32)(35,40)(36,38)(37,39);(1,5,6,2)(3,8,7,4)(9,14,15,22)(10,17,12,20)(11,19,18,24)(13,21,16,23)(25,29,30,27)(26,34,32,31)(28,36,37,35)(33,40,39,38)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=i,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, 9949750);