#8387967

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

4542 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8295163);