#10411706

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

4281 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=h,e^2=h,f^2=gh,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, 10318902);