#10529809

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

4704 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=g,e^2=h,f^2=hi,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, 10437005);