#8901616

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4168 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8808812);