#8254714

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

4128 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)(11,18)(12,13)(14,15)(16,19)(17,22)(20,23)(21,24)(25,26,31,29)(27,32,37,38)(28,34,30,33)(35,40,36,39);(1,2,5,4)(3,7,8,6)(10,15)(11,19)(12,22)(21,23)(25,27)(26,32)(28,35)(29,38)(30,36)(31,37)(33,39)(34,40);(2,4)(6,7)(10,15)(13,17)(16,18)(21,23)(25,28,31,30)(26,33,29,34)(27,35,37,36)(32,39,38,40);(9,11,14,19)(10,16,15,18)(12,20,22,24)(13,21,17,23)(25,29,31,26)(27,32,37,38)(28,34,30,33)(35,39,36,40);(1,3,5,8)(2,6,4,7)(9,12)(10,17)(11,20)(13,15)(14,22)(16,23)(18,21)(19,24)(25,30,31,28)(26,34,29,33)(27,36,37,35)(32,40,38,39);(1,4,5,2)(3,7,8,6)(9,13,14,17)(10,12,15,22)(11,21,19,23)(16,20,18,24)

Polycyclic presentation:

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

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, 8161910);