#8402554

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4262 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=ghi,e^2=h,f^2=gi,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, 8309750);