#10511118

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

4713 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,d^2=g,e^2=i,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, 10418314);