#9815024

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

4318 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9722220);