#10480707

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

4832 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:

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

Polycyclic presentation:

2^9|d^2=gi,e^2=g,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, 10387903);