#10482494

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

4396 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)(11,16)(12,15)(13,17)(14,18)(19,23)(20,21)(22,24)(28,36)(31,37)(34,40)(38,39);(2,3)(6,7)(9,11,15,14)(10,16,12,18)(13,19,21,22)(17,23,20,24)(25,26)(27,32)(28,37)(29,33)(30,35)(31,36)(34,38)(39,40);(1,3,5,2)(4,7,8,6)(9,10,15,12)(11,18,14,16)(13,17,21,20)(19,24,22,23)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,40)(37,39);(1,4,5,8)(2,6,3,7)(9,12,15,10)(11,16,14,18)(13,17,21,20)(19,24,22,23)(25,28,29,36)(26,31,33,37)(27,34,35,40)(30,38,32,39);(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,40)(37,38);(9,14,15,11)(10,16,12,18)(13,19,21,22)(17,24,20,23)(25,26)(27,30)(28,31)(29,33)(32,35)(34,38)(36,37)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=i,f^2=h,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, 10389690);