#10519383

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=ghi,e^2=hi,f^2=gh,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, 10426579);