#9937774

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

4365 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9844970);