#8898102

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

4569 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805298);