#10463946

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

4315 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10371142);