#10411765

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

4508 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)(11,17)(12,19)(13,21)(14,15)(16,23)(18,20)(22,24)(25,26)(27,33)(28,32)(29,31)(30,34)(35,40)(36,39)(37,38);(1,3,7,6)(2,4,8,5)(10,15)(12,17)(16,21)(20,24)(25,27,31,28)(26,32,29,33)(30,35,38,36)(34,39,37,40);(1,4,7,5)(2,6,8,3)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,27,31,28)(26,33,29,32)(30,35,38,36)(34,40,37,39);(1,4,7,5)(2,3,8,6)(9,11)(10,12)(13,22)(14,19)(15,17)(16,24)(18,23)(20,21)(25,28,31,27)(26,33,29,32)(30,35,38,36)(34,39,37,40);(1,5,7,4)(2,3,8,6)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,22)(23,24)(25,29,31,26)(27,32,28,33)(30,37,38,34)(35,39,36,40);(1,6,7,3)(2,4,8,5)(9,13,14,23)(10,16,15,21)(11,18,19,22)(12,20,17,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10318961);