#8716828

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

4386 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)(11,19)(12,21)(13,16)(14,20)(15,17)(18,23)(22,24)(25,26,31,34)(27,32,37,30)(28,29,40,33)(35,39,38,36);(1,2,5,8)(3,4,7,6)(9,11,15,13)(10,16,17,19)(12,22,23,20)(14,18,24,21)(25,27,31,37)(26,32,34,30)(28,38,40,35)(29,36,33,39);(1,3)(2,6)(4,8)(5,7)(10,17)(12,23)(16,19)(20,22)(25,28,31,40)(26,33,34,29)(27,35,37,38)(30,39,32,36);(1,4,5,6)(2,7,8,3)(9,12)(10,18)(11,20)(13,22)(14,16)(15,23)(17,21)(19,24)(25,28)(26,29)(27,35)(30,36)(31,40)(32,39)(33,34)(37,38);(1,3)(2,4)(5,7)(6,8)(9,13,15,11)(10,19,17,16)(12,22,23,20)(14,18,24,21)(25,29)(26,28)(27,36)(30,35)(31,33)(32,38)(34,40)(37,39);(1,4)(2,7)(3,8)(5,6)(9,14,15,24)(10,20,17,22)(11,18,13,21)(12,19,23,16)(25,30)(26,27)(28,39)(29,38)(31,32)(33,35)(34,37)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8624024);