#10188333

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

4525 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10095529);