#8898920

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

4378 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)(3,7)(4,8)(5,6)(9,10,15,18)(11,19,21,16)(12,22,20,17)(13,14,23,24)(26,31)(29,32)(33,38)(37,39);(2,6)(3,8)(9,11)(10,16)(12,13)(14,17)(15,21)(18,19)(20,23)(22,24)(28,36)(33,38)(35,40)(37,39);(1,3)(2,4)(5,8)(6,7)(9,12,15,20)(10,17,18,22)(11,13,21,23)(14,19,24,16)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,4)(2,3)(5,7)(6,8)(9,11)(10,16)(12,13)(14,17)(15,21)(18,19)(20,23)(22,24)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(1,2)(3,4)(5,6)(7,8)(9,13)(10,14)(11,20)(12,21)(15,23)(16,22)(17,19)(18,24)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(1,4)(2,3)(5,7)(6,8)(9,14)(10,13)(11,17)(12,16)(15,24)(18,23)(19,20)(21,22)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=gh,f^2=gh,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, 8806116);