#8418158

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	#42805	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4231 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)(3,5)(4,7)(6,8)(9,10)(11,16)(12,22)(13,17)(14,18)(15,19)(20,24)(21,23)(25,26)(27,35)(28,29)(30,32)(31,34)(33,40)(36,37)(38,39);(2,3)(6,7)(9,11)(10,16)(12,14)(13,23)(15,20)(17,21)(18,22)(19,24)(25,27,31,30)(26,32,34,35)(28,38,33,36)(29,37,40,39);(1,2,5,3)(4,6,8,7)(11,20)(14,21)(16,24)(18,23)(25,28,31,33)(26,29,34,40)(27,36,30,38)(32,39,35,37);(1,3,5,2)(4,7,8,6)(9,12)(10,13)(11,21)(14,20)(15,17)(16,18)(19,22)(23,24)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38);(1,4)(2,6)(3,7)(5,8)(9,13)(10,17)(11,18)(12,19)(14,16)(15,22)(20,23)(21,24)(25,30,31,27)(26,32,34,35)(28,38,33,36)(29,39,40,37);(9,14,15,21)(10,18,19,23)(11,12,20,17)(13,16,22,24)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=h,e^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, 8325354);