#8947429

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

4360 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=i,e^2=h,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, 8854625);