#9938053

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

4363 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)(3,8)(4,6)(5,7)(9,10,15,17)(11,16,12,22)(13,18,23,21)(14,20,24,19)(25,26)(27,28)(29,32)(30,38)(31,33)(34,39)(35,40)(36,37);(1,3,5,6)(2,4,7,8)(9,11)(10,16)(12,15)(13,20)(14,21)(17,22)(18,24)(19,23)(25,27,31,29)(26,32,33,28)(30,35,39,37)(34,40,38,36);(10,17)(11,12)(18,21)(19,20)(26,33)(28,32)(34,38)(36,40);(1,2)(3,8)(4,6)(5,7)(9,12,15,11)(10,16,17,22)(13,19,23,20)(14,18,24,21)(25,28,31,32)(26,27,33,29)(30,36,39,40)(34,35,38,37);(1,4,5,8)(2,6,7,3)(9,13)(10,18)(11,20)(12,19)(14,22)(15,23)(16,24)(17,21)(25,29,31,27)(26,32,33,28)(30,35,39,37)(34,36,38,40);(1,2)(3,4)(5,7)(6,8)(9,14,15,24)(10,19,17,20)(11,21,12,18)(13,16,23,22)(25,30,31,39)(26,34,33,38)(27,35,29,37)(28,36,32,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=h,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, 9845249);