#8418648

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×#535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4372 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8325844);