#10451677

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

4673 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)(10,14)(12,20)(15,17)(18,21)(25,26)(27,32)(28,30)(29,36)(31,35)(33,37)(34,38)(39,40);(10,14)(15,17)(16,22)(23,24)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,26,30,28)(27,32,35,31)(29,33,37,36)(34,39,40,38);(3,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,28,30,26)(27,32,35,31)(29,36,37,33)(34,39,40,38);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,20)(22,23)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,18)(13,20)(14,22)(15,23)(17,24)(19,21)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)

Polycyclic presentation:

2^9|c^2=g,d^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, 10358873);