#10521356

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

4775 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=ghi,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, 10428552);