#10519371

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

4432 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)(11,16)(12,17)(13,18)(14,15)(19,21)(20,22)(23,24)(25,26)(27,31)(28,38)(29,33)(30,34)(32,37)(35,40)(36,39);(10,15)(11,19)(17,20)(18,24)(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)(9,11,14,19)(10,16,15,21)(12,18,22,24)(13,20,23,17)(25,28,30,37)(26,32,34,38)(27,36,35,29)(31,33,40,39);(1,4,6,2)(3,8,5,7)(9,12)(10,17)(11,18)(13,16)(14,22)(15,20)(19,24)(21,23)(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)(10,18)(11,20)(12,21)(14,23)(15,24)(16,22)(17,19)(25,26)(27,31)(28,32)(29,39)(30,34)(33,36)(35,40)(37,38)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=ghi,e^2=hi,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, 10426567);