#10534396

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

4375 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=h,f^2=ghi,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, 10441592);