#10247288

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

4565 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154484);