#8109960

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

4284 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=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, 8017156);