#9909808

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

4688 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)(3,7)(4,6)(5,8)(9,10,14,16)(11,18,22,15)(12,21,23,17)(13,20,24,19)(25,26,31,28)(27,32,29,33)(30,34,38,36)(35,39,37,40);(1,2,4,6)(3,5,8,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,18)(17,24)(19,23)(25,26,31,28)(27,33,29,32)(30,34,38,36)(35,40,37,39);(1,2)(3,5)(4,6)(7,8)(10,16)(12,23)(15,18)(19,20)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,20)(13,18)(14,23)(15,24)(16,21)(19,22)(25,28,31,26)(27,33,29,32)(30,36,38,34)(35,40,37,39);(9,11)(10,18)(12,20)(13,17)(14,22)(15,16)(19,23)(21,24)(25,29,31,27)(26,33,28,32)(30,35,38,37)(34,39,36,40);(9,13)(10,19)(11,21)(12,18)(14,24)(15,23)(16,20)(17,22)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=ghi,d^2=ghi,e^2=ghi,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, 9817004);