#9909741

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

4303 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)(11,15)(12,16)(13,21)(14,18)(17,23)(19,20)(22,24);(1,3,5,8)(2,6,7,4)(9,11,14,16)(10,15,18,12)(13,19,23,24)(17,22,21,20)(25,26,31,28)(27,33,29,32)(30,34,38,36)(35,40,37,39);(1,3)(2,4)(5,8)(6,7)(9,10,14,18)(11,12,16,15)(13,17,23,21)(19,20,24,22)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(1,4)(2,3)(5,6)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,24)(21,23)(25,28,31,26)(27,33,29,32)(30,36,38,34)(35,40,37,39);(9,12,14,15)(10,16,18,11)(13,22,23,20)(17,19,21,24)(25,29,31,27)(26,33,28,32)(30,35,38,37)(34,39,36,40);(9,13)(10,17)(11,19)(12,20)(14,23)(15,22)(16,24)(18,21)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=ghi,e^2=gi,f^2=ghi,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, 9816937);