#8341781

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

4135 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)(11,19)(12,14)(13,17)(15,16)(18,22)(20,21)(23,24)(27,33)(30,37)(34,39)(38,40);(1,2,5,4)(3,6,7,8)(10,16)(12,22)(17,19)(20,23)(25,26,29,32)(27,30,33,37)(28,35,36,31)(34,40,39,38);(2,4)(6,8)(9,11,15,13)(10,17,16,19)(12,20,22,23)(14,24,18,21)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(3,7)(6,8)(9,12)(10,18)(11,20)(13,23)(14,16)(15,22)(17,21)(19,24)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40);(1,3,5,7)(2,6,4,8)(9,13,15,11)(10,19,16,17)(12,20,22,23)(14,24,18,21)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(1,4,5,2)(3,8,7,6)(9,14,15,18)(10,12,16,22)(11,21,13,24)(17,23,19,20)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=gh,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8248977);