#10041407

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

4438 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)(9,10,14,11)(12,15,19,17)(13,16,22,18)(20,23,21,24)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,39,38);(1,3,5,8)(2,4,6,7)(10,11)(15,17)(16,18)(23,24)(26,28)(27,33)(32,36)(35,39);(2,6)(3,8)(12,19)(15,17)(20,21)(23,24)(26,28)(27,33)(32,36)(35,39);(1,3)(2,7)(4,6)(5,8)(9,11,14,10)(12,17,19,15)(13,18,22,16)(20,24,21,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3)(2,4)(5,8)(6,7)(9,12,14,19)(10,15,11,17)(13,21,22,20)(16,24,18,23)(25,28,30,26)(27,34,33,31)(29,32,37,36)(35,38,39,40);(1,4)(2,8)(3,6)(5,7)(9,13,14,22)(10,16,11,18)(12,20,19,21)(15,23,17,24)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=ghi,e^2=g,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, 9948603);