#10368338

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

4675 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10275534);