#8932410

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

4250 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839606);