#10511099

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

4472 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)(11,15)(12,21)(13,18)(14,17)(16,19)(20,24)(22,23)(25,26)(27,32)(28,30)(29,36)(31,35)(33,37)(34,38)(39,40);(9,11,16,14)(10,17,19,15)(12,20,13,22)(18,24,21,23)(25,26,30,28)(27,32,35,31)(29,33,37,36)(34,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,18,19,21)(11,20,14,22)(15,23,17,24)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(3,7)(6,8)(9,13,16,12)(10,18,19,21)(11,20,14,22)(15,24,17,23)(25,28,30,26)(27,32,35,31)(29,36,37,33)(34,39,40,38);(1,3)(2,6)(4,7)(5,8)(9,14,16,11)(10,17,19,15)(12,22,13,20)(18,24,21,23)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,15,16,17)(10,14,19,11)(12,23,13,24)(18,22,21,20)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=g,e^2=gi,f^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, 10418295);