#8935274

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

4336 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8842470);