#8914373

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

4313 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=g,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8821569);