#9923669

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

4608 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)(11,17)(12,16)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26)(27,33)(28,37)(29,31)(30,34)(32,35)(36,38)(39,40);(1,3,5,6)(2,4,7,8)(9,11)(10,13)(12,20)(14,18)(15,17)(16,24)(19,22)(21,23)(25,27,32,29)(26,33,35,31)(28,36,30,39)(34,40,37,38);(10,15)(11,18)(16,23)(19,20)(25,28,32,30)(26,34,35,37)(27,36,29,39)(31,38,33,40);(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,22)(23,24)(25,29,32,27)(26,31,35,33)(28,36,30,39)(34,40,37,38);(1,4,5,8)(2,6,7,3)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,30,32,28)(26,34,35,37)(27,36,29,39)(31,40,33,38);(1,2)(3,4)(5,7)(6,8)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,31)(26,27)(28,38)(29,35)(30,40)(32,33)(34,36)(37,39)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=hi,e^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9830865);