#10408084

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

4729 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10315280);